1MA 06. LLOCS GEOMÈTRICS/1MA.06.1 LíniesPuntsNotabTriang 1MA.06.1.10ADT BISECTRIU ANGLE Bisectriu

La bisectriu és el conjunt de punts equidistants

dels dos segments que determinen un angle.

Divideix l'angle

en dos angles iguals.

Les 3 bisectrius

es tallen en un punt anomenat incentre (centre de la circumferència inscrita).

]]> 0.0000000 0.0000000 0 1MA.06.1.10BDT CÀLCUL EQUACIÓ BISECTRIUS Com es calcula l'equació de la bisectriu TOTS ELS PUNTS de la bisectriu de dues rectes es troben a la mateixa distància de les dues rectes. Exemple de càlcul:
La bisectriu de les rectes 2x + y - 4 = 0 i x - 2y + 9 = 0 es calcula igualant les distancies:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#003300¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»y«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»4«/mn»«/mrow»«/mfenced»«msqrt»«mn mathvariant=¨bold¨»5«/mn»«/msqrt»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»2«/mn»«mi mathvariant=¨bold¨»y«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»9«/mn»«/mrow»«/mfenced»«msqrt»«mn mathvariant=¨bold¨»5«/mn»«/msqrt»«/mfrac»«/mrow»«/mstyle»«/math»

Igualem els denominadors amb el mcm i els suprimim.
En treure els valors absoluts ens queden dues equacions:

Si tot té el mateix signe: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»4«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»9«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8660;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»3«/mn»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»13«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»0«/mn»«/mrow»«/mstyle»«/math»
Si tenen signes oposats: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»4«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»9«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8660;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»3«/mn»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»5«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»0«/mn»«/mrow»«/mstyle»«/math»

]]> 0.0000000 0.0000000 0 1MA.06.1.11Q BisectriuAngle Es defineix la bisectriu d'un angle com el conjunt de punts equidistants dels segments que determinen l'angle.

En el triangle de vèrtex «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mfenced mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mfenced mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»C«/mi»«mfenced mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»c«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»c«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math», quina és l'equació explícita de la bisectriu de l'angle A?

]]> Gràficament: #G1

]]> 1.0000000 0.3333333 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>enunciat</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c2</mi><mo>=</mo><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c1</mi><mo>,</mo><mi>c2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d1</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d2</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d3</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>AB</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>b1</mi><mo>-</mo><mi>a1</mi><mo>,</mo><mi>b2</mi><mo>-</mo><mi>a2</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mi>AC</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>c1</mi><mo>-</mo><mi>a1</mi><mo>,</mo><mi>c2</mi><mo>-</mo><mi>a2</mi></mrow></mfenced></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>d1</mi><mo><</mo><mi>d2</mi><mo>+</mo><mi>d3</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>d2</mi><mo><</mo><mi>d1</mi><mo>+</mo><mi>d3</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>d3</mi><mo><</mo><mi>d1</mi><mo>+</mo><mi>d2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mfenced close="‖" open="‖"><mi>AB</mi></mfenced><mo>∈</mo><rationals/></mrow></mfenced><mo>∧</mo><mfenced><mrow><mfenced close="‖" open="‖"><mi>AC</mi></mfenced><mo>∈</mo><rationals/></mrow></mfenced></mrow></apply></math></input></command><command><input><math 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align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mo> </mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>4</mn><mo>,</mo><mn>3</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mn>3</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mn>4</mn><mo>,</mo><mo>-</mo><mn>2</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AB</mi><mo>,</mo><mo> </mo><mi>AC</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>8</mn><mo>,</mo><mn>0</mn></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>5</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Primer calcules l'equació CARTESIANA (de la forma ax+ by + c = 0) de la recta AB que és una recta que passa pel punt A i que té, per vector director, el vector AB.

Després calcules l'equació CARTESIANA de la recta AC que passa pel punt A i que té per vector director, el vector AC.

Agafes un punt qualsevol P(x,y), i escrius la seva distància amb l'expressió

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»d«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»,«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»r«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mfrac mathcolor=¨#0000FF¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«msub»«mi mathvariant=¨bold¨»ax«/mi»«mi mathvariant=¨bold¨»P«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»by«/mi»«mi mathvariant=¨bold¨»P«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»c«/mi»«/mrow»«/mfenced»«msqrt»«msup»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mfrac»«/mstyle»«/math»

a les dues rectes AB i AC i les iguales

Atenció, com que són distàncies, apareixen valors absoluts. Cal fixar-se en el gràfic per escollir la bisectriu de l'angle i no la seva perpendicular.

]]> 1MA.06.1.12Q BisectriuRectes La bisectriu de 2 rectes que es tallen és una recta, els punts de la qual són equidistants de les 2 rectes. Utilitza el concepte de distància d'un punt a una recta per resoldre.

Troba l'equació de les bisectrius de les dues rectes d'equacions «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»e«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»e«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«/mrow»«/mstyle»«/math»

Format de la resposta:
escriviu les dues equacions cartesianes:
{2x-14y+7=0, 14x+2y-5=0} ]]> Bisectrius en blau

#G1

]]> 1.0000000 0.3333333 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>mr</mi><mo>,</mo><mi>ms</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>,</mo><mn>5</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Cal igualar les distàncies

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#0000FF¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»w«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«msqrt»«msup»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»vr«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»vr«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»w«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«msqrt»«msup»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»vs«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»vs«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mfrac»«/mrow»«/mstyle»«/math»

i treure els denominadors i el valor absolut

]]> Si es treuen denominadors

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfrac mathcolor=¨#0000FF¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»w«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«msqrt»«msup»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»vr«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»vr«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»w«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«msqrt»«msup»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»vs«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»vs«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8660;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»11«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨|¨ close=¨|¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»w«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»12«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfenced mathcolor=¨#0000FF¨ open=¨|¨ close=¨|¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»w«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math»

I, traient valors absoluts s'obtenen les DUES EQUACIONS

]]> 1MA.06.1.20DT ALTURA TRIANGLE Altura

Una altura és una recta que passa per un vèrtex i que és perpendicular al costat oposat.

Les 3 altures es tallen en l'ortocentre.

]]> 0.0000000 0.0000000 0 1MA.06.1.21Q AlturaTriangle Es defineix l'altura en un triangle com la recta perpendicular a un costat traçada des del vèrtex oposat.

]]> Gràficament: #G1

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</mo><mi>amplada_línia</mi><mo>=</mo><mn>4</mn><mo>,</mo><mi>mostrar_etiqueta</mi><mo>=</mo><mi>cert</mi></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>s</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>cian</mi><mo>,</mo><mo> </mo><mi>amplada_línia</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><mn>3</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>3</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BC</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mn>3</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>vp</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>*</mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> L'altura passa pel punt A, només necessitem és el seu vector director.

Com que aquesta altura és perpendicular al segment BC, el seu vector director és perpendicular al vector BC.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mover mathcolor=¨#000066¨»«mi mathvariant=¨bold¨»BC«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»BC«/mi»«/mrow»«/mstyle»«/math» i un vector perpendicular al segment BC és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»vp«/mi»«/mrow»«/mstyle»«/math»

]]> 1MA.06.1.22Q Determinar l'ortocentre

L'altura d'un triangle és la recta perpendicular a un costat traçada des del vèrtex oposat.

L'ortocentre és el punt d'intersecció de les altures que corresponen als 3 vèrtex.

a) Quina és l'equació explícita de l'altura traçada des del vèrtex A?

b) Quina és l'equació explícita de l'altura traçada des del vèrtex B?

c) Quines són les coordenades de l'ortocentre del triangle? Format (2,3)

]]> Gràficament: #G1

]]> 1.0000000 0.3333333 0 0 a) =#sol1b) =#sol2c) =#sol3]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>enunciat</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable 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linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question>

Com l'altura passa per A el que necessitem és el seu vector director; i com que aquesta altura és perpendicular al segment BC, el seu vector director és perpendicular al vector BC.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mover mathcolor=¨#000066¨»«mi mathvariant=¨bold¨»BC«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»BC«/mi»«/mrow»«/mstyle»«/math» i un vector perpendicular al segment BC és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»PBC«/mi»«/mstyle»«/math»

Fem el mateix per l'altura que passa per B, amb el vector perpendicular a AC: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»PAC«/mi»«/mrow»«/mstyle»«/math»

Gràficament: #G1

]]> Resolem per igualació el sistema d'equacions format per «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»sol«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»sol«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»2«/mn»«/mrow»«/mstyle»«/math» per trobar el seu punt d'intersecció per determinar l'ortocentre.

]]> 1MA.06.1.30DT MITJANES TRIANGLE

Mitjanes

La mitjana és la recta que passa per un vèrtex i el punt mitjà del costat oposat.

Les tres mitjanes es tallen en el baricentre.

]]> 0.0000000 0.0000000 0 1MA.06.1.31Q Equació mitjana d'un triangle Es defineix la mitjana en un triangle com la recta que uneix un vèrtex al punt mitjà del costat oposat.

]]> Gràficament: #G1

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>enunciat</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c1</mi><mo>,</mo><mi>c2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d1</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d2</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d3</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>s</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>d1</mi><mo><</mo><mi>d2</mi><mo>+</mo><mi>d3</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>d2</mi><mo><</mo><mi>d1</mi><mo>+</mo><mi>d3</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>d3</mi><mo><</mo><mi>d1</mi><mo>+</mo><mi>d2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>></mo><mn>3</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>A</mi><mo>≠</mo><mi>B</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>B</mi><mo>≠</mo><mi>C</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>A</mi><mo>≠</mo><mi>C</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math 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Es calcula el punt mitjà del segment BC: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»M«/mi»«/mrow»«/mstyle»«/math»

Un vector director de la mitjana és el vector que va de A a M : «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»vp«/mi»«/mrow»«/mstyle»«/math»

Gràficament: #G1

]]> 1MA.06.1.32Q Determinar el baricentre La mitjana en un triangle és la recta que uneix un vèrtex al punt mitjà del costat oposat.

El baricentre és el punt d'intersecció de les 3 mitjanes.

a) Calcula l'equació explícita de la mitjana traçada des del vèrtex A

b) Calcula l'equació explícita de la mitjana traçada des del vèrtex B

c) Determina les coordenades del baricentre. Format: (2,3)

]]> Gràficament: #G1

]]> 1.0000000 0.3333333 0 0 a) =#sol1b) =#sol2c) =#sol3]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>enunciat</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c1</mi><mo>,</mo><mi>c2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d1</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d2</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d3</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>s</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>></mo><mn>3</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>A</mi><mo>≠</mo><mi>B</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>A</mi><mo>≠</mo><mi>C</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>C</mi><mo>≠</mo><mi>B</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math 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open="["><mrow><mfrac><mrow><mi>b1</mi><mo>+</mo><mi>c1</mi></mrow><mn>2</mn></mfrac><mo>-</mo><mi>a1</mi><mo>,</mo><mfrac><mrow><mi>b2</mi><mo>+</mo><mi>c2</mi></mrow><mn>2</mn></mfrac><mo>-</mo><mi>a2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M2</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mfrac><mrow><mi>a1</mi><mo>+</mo><mi>c1</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mfrac><mrow><mi>a2</mi><mo>+</mo><mi>c2</mi></mrow><mn>2</mn></mfrac><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>vp2</mi><mo>=</mo><mfenced close="]" open="["><mrow><mfrac><mrow><mi>a1</mi><mo>+</mo><mi>c1</mi></mrow><mn>2</mn></mfrac><mo>-</mo><mi>b1</mi><mo>,</mo><mfrac><mrow><mi>a2</mi><mo>+</mo><mi>c2</mi></mrow><mn>2</mn></mfrac><mo>-</mo><mi>b2</mi></mrow></mfenced></math></input></command><command><input><math 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Es calcula el punt mitjà M1 del segment BC: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»M«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«/mstyle»«/math»

Un vector director de la mitjana és el vector que va de A a M1 : «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»vp«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#000066¨»1«/mn»«/mstyle»«/math»

Es calcula el punt mitjà M2 del segment AC: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»M«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»2«/mn»«/mstyle»«/math»

Un vector director de la mitjana és el vector que va de B a M2 : «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»vp«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#000066¨»2«/mn»«/mstyle»«/math»

Gràficament: #G1

]]> El baricentre es troba resolent, per igualació, el sistema format per les equacions de les mitjanes: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»sol«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»sol«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»2«/mn»«/mrow»«/mstyle»«/math»

]]> 1MA.06.1.40DT MEDIATRIU TRIANGLE Mediatriu

Una mediatriu és una recta perpendicular a un costat que passa pel seu punt mitjà.

Les tres mediatrius es tallen el el circumcentre

(centre de la circumferència que passa pels 3 vèrtex)

]]> 0.0000000 0.0000000 0 1MA.06.1.41Q Equació mediatriu segment La mediatriu d'un segment és la perpendicular al segment que passa pel seu punt mitjà,

Troba l'equació explícita de la mediatriu del segment AB amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«/mstyle»«/math».

]]> Gràficament: #G1

]]> 1.0000000 0.3333333 0 0 #sol1]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mi style="color:#ffc800">variables</mi><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>v1</mi><mo>=</mo><mi>b1</mi><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>v2</mi><mo>=</mo><mi>b2</mi><mo>-</mo><mi>a2</mi></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>a1</mi><mo>≠</mo><mi>b1</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>a2</mi><mo>≠</mo><mi>b2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>distància</mi><mfenced><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></mfenced><mo>></mo><mn>4</mn></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math 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align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>mostrar_etiqueta</mi><mo>=</mo><mi>cert</mi><mo>,</mo><mi>midaPunt</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>r</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>mostrar_etiqueta</mi><mo>=</mo><mi>fals</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>s</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>mostrar_etiqueta</mi><mo>=</mo><mi>fals</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>M</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mn>4</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m1</mi><mo>,</mo><mi>m2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>,</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>*</mo><mi>x</mi><mo>+</mo><mn>9</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>vr</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo>-</mo><mn>5</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>vp</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mn>5</mn><mo>,</mo><mo>-</mo><mn>2</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mo>*</mo><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>10</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> El segment AB és el segment vermell: #G1

Primer es determina el punt mitjà de AB, pensant que M és l'extrem d'un vector «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mover mathcolor=¨#000066¨»«mi mathvariant=¨bold¨»AM«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mstyle»«/math» tal que «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mover mathcolor=¨#000066¨»«mi mathvariant=¨bold¨»AM«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mfrac mathcolor=¨#000066¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mover mathcolor=¨#000066¨»«mi mathvariant=¨bold¨»AB«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mfrac mathcolor=¨#000066¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»vr«/mi»«/mstyle»«/math».

I el punt M que és l'extrem ("extrem=origen+vector) es calcula amb:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»M«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»:«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mfrac mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»vr«/mi»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math»

]]> Un cop es té el punt M «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»M«/mi»«/mrow»«/mstyle»«/math», cal trobar el vector director de la perpendicular.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»vr«/mi»«/mrow»«/mstyle»«/math» és director de AB.

Un vector perpendicular a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»vr«/mi»«/mrow»«/mstyle»«/math» és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»vp«/mi»«/mrow»«/mstyle»«/math», que és el vector director de la recta demanada.

Tenim el punt, M«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»M«/mi»«/mrow»«/mstyle»«/math», i el vector, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»vp«/mi»«/mrow»«/mstyle»«/math». Podem trobar l'equació de la recta.

]]> 1MA.06.1.42Q Circumcentre La mediatriu d'un segment és la perpendicular al segment que passa pel seu punt mitjà.

El circumcentre d'un triangle és el punt d'intersecció de les mediatrius dels 3 costats.

En el triangle ABC amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mfenced mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»c«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»c«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math», determina:

a) l'equació explícita de la mediatriu del costat BC

b) l'equació explícita de la mediatriu del costat AC

c) les coordenades del circumcentre Format: (1,-2)

]]> Gràficament: #G1

]]> 1.0000000 0.3333333 0 0 a) =#sol1b) =#sol2c) =#sol3]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>enunciat</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c1</mi><mo>,</mo><mi>c2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d1</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d2</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d3</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>s</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>d1</mi><mo><</mo><mi>d2</mi><mo>+</mo><mi>d3</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>d2</mi><mo><</mo><mi>d1</mi><mo>+</mo><mi>d3</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>d3</mi><mo><</mo><mi>d1</mi><mo>+</mo><mi>d2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>></mo><mn>3</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>A</mi><mo>≠</mo><mi>B</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>A</mi><mo>≠</mo><mi>C</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>B</mi><mo>≠</mo><mi>C</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mi>triangle</mi><mo>(</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo><mo>,</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo><mo>,</mo><mi>punt</mi><mo>(</mo><mi>c1</mi><mo>,</mo><mi>c2</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>resolució</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>mediatriu</mi><mo>(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>mediatriu</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mi>circumcentre</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M1</mi><mo>=</mo><mi>punt</mi><mfenced><mrow><mfrac><mrow><mi>b1</mi><mo>+</mo><mi>c1</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mfrac><mrow><mi>b2</mi><mo>+</mo><mi>c2</mi></mrow><mn>2</mn></mfrac></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M2</mi><mo>=</mo><mi>punt</mi><mfenced><mrow><mfrac><mrow><mi>a1</mi><mo>+</mo><mi>c1</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mfrac><mrow><mi>a2</mi><mo>+</mo><mi>c2</mi></mrow><mn>2</mn></mfrac></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BC</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>c1</mi><mo>-</mo><mi>b1</mi><mo>,</mo><mi>c2</mi><mo>-</mo><mi>b2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PBC</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>b2</mi><mo>-</mo><mi>c2</mi><mo>,</mo><mi>c1</mi><mo>-</mo><mi>b1</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AC</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>c1</mi><mo>-</mo><mi>a1</mi><mo>,</mo><mi>c2</mi><mo>-</mo><mi>a2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PAC</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>a2</mi><mo>-</mo><mi>c2</mi><mo>,</mo><mi>c1</mi><mo>-</mo><mi>a1</mi></mrow></mfenced></math></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Gràfic</mi></math></comment></maction><command><input><math 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Per calcular les mediatrius, es troben les coordenades dels punts mitjans,

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»M«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»M«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»2«/mn»«/mrow»«/mstyle»«/math».

El vector director de la mediatriu de BC és el vector perpendicular a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»BC«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»:«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»PBC«/mi»«/mrow»«/mstyle»«/math»

El vector director de la mediatriu de AC és el vector perpendicular a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»AC«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»:«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»PAC«/mi»«/mrow»«/mstyle»«/math»

Gràficament: #G1

]]> Per trobar el circumcentre, n'hi ha prou amb resoldre el sistema de les dues equacions de les mediatrius, per igualació:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced mathcolor=¨#00007F¨ open=¨{¨ close=¨¨»«mtable columnalign=¨left¨»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»sol«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»sol«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«/mstyle»«/math»

]]> 1MA.06.1.81 OBERTA Baricentre Els vèrtex d'un triangle són els punts A(4,1), B(0,3) i C(6,3)

a) Determina les coordenades dels punts mitjans dels segments, AB, AC i BC.

b) Una mitjana és una recta que passa per un vèrtex d'un triangle i pel punt mitjà del costat oposat a aquest vèrtex. Troba les equacions de les mitjanes que passen per A per B i per C.

c) El baricentre és el punt on es tallen les mitjanes. Demostra que el baricentre del triangle té per coordenades: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced mathcolor=¨#003300¨»«mrow»«mfrac»«mrow»«msub»«mi mathvariant=¨bold¨»x«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/mrow»«mn mathvariant=¨bold¨»3«/mn»«/mfrac»«mo mathvariant=¨bold¨»,«/mo»«mfrac»«mrow»«msub»«mi mathvariant=¨bold¨»y«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»y«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»y«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/mrow»«mn mathvariant=¨bold¨»3«/mn»«/mfrac»«/mrow»«/mfenced»«/mstyle»«/math» si x_A, y_A són les coordenades de A, x_B, y_B les de B i x_C, y_C les de C

]]> 1.0000000 0.0000000 0 editor 1 15 0 0 1MA.06.1.82 OBERTA Triangle equilàter: altura i àrea NO arrodoneixis les quantitats. Treballa amb arrels si cal.

Un triangle equilàter ABC té dos vèrtex en A(4,6) i B(4,-2).

a) Calcula les coordenades del 3r vèrtex C. Quantes solucions hi ha? Comprova-ho gràficament.

En els apartats següents, fes els càlculs amb el valor arrodonit C(-3,2).

b) Determina l'equació de l'altura traçada des del vèrtex A (recta perpendicular al costat BC que passa per A).

c) Quant mesura l'angle B?

d) Calcula l'àrea del triangle.

]]> 1.0000000 0.0000000 0 editor 1 15 0 0 1MA 06. LLOCS GEOMÈTRICS/1MA.06.2 Paral·lelogSimetries 1MA.06.2.10DT PARAL·LELOGRAM

Paral·lelogram

Té els costats paral·lels iguals i els angles oposats iguals. Dos angles adjacents sumen 180º.

Perímetre:P= suma dels 4 costats.

Àrea: S= b· h

Les diagonals D i d es tallen en el seu punt mitjà.

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 0.0000000 0.0000000 0 1MA.06.2.11Q EqCostatParal·lelogram Considera els punts «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»C«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math». A partir d'aquests punts, es pot construir el paral·lelogram ABCD.

Escriu les equacions de les rectes que corresponen als costats AB i CD.

Format:

AB= paramètriques {x=k+1,y=k-1}

CD=explícita

]]> #G2

]]> 1.0000000 0.3333333 0 0 AB=#sol1CD=#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mi style="color:#ffc800">variables</mi><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable 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align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>4</mn><mo>*</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>4</mn><mo>*</mo><mi>k</mi><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler2</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>C</mi><mi>D</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="inputCompound">true</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> a) L'equació de la recta AB és l'equació d'una recta que passa pel punt A «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math» i que té per vector director el vector «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mover mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨»AB«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»vr«/mi»«/mstyle»«/math» .

La recta és la recta vermella: #G1

]]> b) La recta que passa per C i D és una recta que passa per C(#c1,#c2) i que és paral·lela a la recta AB. Es pot doncs emprar, com a vector director, el vector «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mover mathcolor=¨#000066¨»«mi mathvariant=¨bold¨»AB«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»vr«/mi»«/mstyle»«/math».

La recta és la recta blava que passa per C: #G2

]]> 1MA.06.2.12Q AnglesParal·lelogram Els 3 vèrtex del paral·lelogram ABCD són «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»C«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math».

Determina quant mesuren els angles A i B

Format:

A= arrodonit en graus: 25º

B = arrodonit en graus: 25º

]]> 1.0000000 0.3333333 0 0 A=#sol1B=#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mi style="color:#ffc800">variables</mi><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mo>-</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c1</mi><mo>,</mo><mi>c2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>v1</mi><mo>=</mo><mi>b1</mi><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>v2</mi><mo>=</mo><mi>b2</mi><mo>-</mo><mi>a2</mi></mtd></mtr><mtr><mtd><mi>u1</mi><mo>=</mo><mi>c1</mi><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>u2</mi><mo>=</mo><mi>c2</mi><mo>-</mo><mi>a2</mi></mtd></mtr><mtr><mtd><mi>r2</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>A</mi><mo>,</mo><mfenced close="]" open="["><mrow><mi>v1</mi><mo>,</mo><mi>v2</mi></mrow></mfenced><mo>)</mo></mtd></mtr></mtable><mrow><mfenced><mrow><mi>v1</mi><mo>*</mo><mi>u1</mi><mo>≠</mo><mn>0</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>distància</mi><mfenced><mrow><mi>C</mi><mo>,</mo><mi>r2</mi></mrow></mfenced><mo>></mo><mn>3</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>u1</mi><mo>*</mo><mi>v2</mi><mo>≠</mo><mi>u2</mi><mo>*</mo><mi>v1</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>c1</mi><mo>-</mo><mi>v1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi><mo>=</mo><mi>c2</mi><mo>-</mo><mi>v2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>punt</mi><mfenced><mrow><mi>d1</mi><mo>,</mo><mi>d2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AB</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>v1</mi><mo>,</mo><mi>v2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AD</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>d1</mi><mo>-</mo><mi>a1</mi><mo>,</mo><mi>d2</mi><mo>-</mo><mi>a2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BA1</mi><mo>=</mo><mi>a1</mi><mo>-</mo><mi>b1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BA2</mi><mo>=</mo><mi>a2</mi><mo>-</mo><mi>b2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BC1</mi><mo>=</mo><mi>c1</mi><mo>-</mo><mi>a1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BC2</mi><mo>=</mo><mi>c2</mi><mo>-</mo><mi>a2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A1</mi><mo>=</mo><mi>angle</mi><mo>(</mo><mi>AB</mi><mo>,</mo><mi>AD</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol11</mi><mo>=</mo><mi>arrodoneix</mi><mfenced><mrow><mi>A1</mi><mo>*</mo><mfrac><mn>180</mn><pi/></mfrac></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol21</mi><mo>=</mo><mn>180</mn><mo>-</mo><mi>sol11</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>sol11</mi><csymbol definitionURL="http://.../units/degree/angular">°</csymbol></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>sol21</mi><csymbol definitionURL="http://.../units/degree/angular">°</csymbol></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T2</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>amplada</mi><mo>=</mo><mn>25</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>25</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>A</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>mostrar_etiqueta</mi><mo>=</mo><mi>cert</mi><mo>,</mo><mi>midaPunt</mi><mo>=</mo><mn>7</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>B</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>mostrar_etiqueta</mi><mo>=</mo><mi>cert</mi><mo>,</mo><mi>midaPunt</mi><mo>=</mo><mn>7</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>C</mi><mo>,</mo><mfenced close="}" open="{"><mtable 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align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>segment</mi><mo>(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>segment</mi><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>AB</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AD</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mn>9</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2.4669</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>141</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>39</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>B</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="inputCompound">true</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> La situació és #G2.

Com que es tracta d'un paral·lelogram, l'angle B és l'angle que determinen els dos vectors «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mover mathcolor=¨#000066¨»«mrow»«mi mathvariant=¨bold¨»BA«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo»§#160;«/mo»«mo mathvariant=¨bold¨»=«/mo»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»BA«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»BA«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mover mathcolor=¨#000066¨»«mi mathvariant=¨bold¨»BC«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mfenced mathcolor=¨#000066¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»B«/mi»«mi mathvariant=¨bold¨»C«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»B«/mi»«mi mathvariant=¨bold¨»C«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math»

L'angle és:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mover mathcolor=¨#00007F¨»«mfenced»«mrow»«mover»«mi mathvariant=¨bold¨»BA«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨»,«/mo»«mover»«mi mathvariant=¨bold¨»BC«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»^«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»acos«/mi»«mfrac mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»BA«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»BC«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»BA«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»BC«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«msqrt»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»BA«/mi»«msup»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»BA«/mi»«msup»«mn mathvariant=¨bold¨»2«/mn»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«mo mathvariant=¨bold¨»§#183;«/mo»«msqrt»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»BC«/mi»«msup»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»BC«/mi»«msup»«mn mathvariant=¨bold¨»2«/mn»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»

]]> L'angle A, en un paral·lelogram és 180º - B

]]> 1MA.06.2.13Q 4t vèrtex i angles paral·lelogram Els 3 vèrtex del paral·lelogram ABCD són «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»C«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math».

Determina el quart vèrtex D i quant mesuren els angles A i B

Format:

D=(1,2)

A= arrodonit en graus: 25º

B = arrodonit en graus: 25º

]]> 1.0000000 0.3333333 0 0 D=#sol3A=#sol1B=#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mi style="color:#ffc800">variables</mi><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mo>-</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c1</mi><mo>,</mo><mi>c2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>v1</mi><mo>=</mo><mi>b1</mi><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>v2</mi><mo>=</mo><mi>b2</mi><mo>-</mo><mi>a2</mi></mtd></mtr><mtr><mtd><mi>u1</mi><mo>=</mo><mi>c1</mi><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>u2</mi><mo>=</mo><mi>c2</mi><mo>-</mo><mi>a2</mi></mtd></mtr><mtr><mtd><mi>r2</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>A</mi><mo>,</mo><mfenced close="]" open="["><mrow><mi>v1</mi><mo>,</mo><mi>v2</mi></mrow></mfenced><mo>)</mo></mtd></mtr></mtable><mrow><mfenced><mrow><mi>v1</mi><mo>*</mo><mi>u1</mi><mo>≠</mo><mn>0</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>distància</mi><mfenced><mrow><mi>C</mi><mo>,</mo><mi>r2</mi></mrow></mfenced><mo>></mo><mn>3</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>u1</mi><mo>*</mo><mi>v2</mi><mo>≠</mo><mi>u2</mi><mo>*</mo><mi>v1</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>c1</mi><mo>-</mo><mi>v1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi><mo>=</mo><mi>c2</mi><mo>-</mo><mi>v2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>punt</mi><mfenced><mrow><mi>d1</mi><mo>,</mo><mi>d2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AB</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>v1</mi><mo>,</mo><mi>v2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BA</mi><mo>=</mo><mo>-</mo><mi>AB</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BC</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>c1</mi><mo>-</mo><mi>b1</mi><mo>,</mo><mi>c2</mi><mo>-</mo><mi>b2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AD</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>d1</mi><mo>-</mo><mi>a1</mi><mo>,</mo><mi>d2</mi><mo>-</mo><mi>a2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BA1</mi><mo>=</mo><mi>a1</mi><mo>-</mo><mi>b1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BA2</mi><mo>=</mo><mi>a2</mi><mo>-</mo><mi>b2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BC1</mi><mo>=</mo><mi>c1</mi><mo>-</mo><mi>b1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BC2</mi><mo>=</mo><mi>c2</mi><mo>-</mo><mi>b2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A1</mi><mo>=</mo><mi>angle</mi><mo>(</mo><mi>AB</mi><mo>,</mo><mi>AD</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol11</mi><mo>=</mo><mi>arrodoneix</mi><mfenced><mrow><mi>A1</mi><mo>*</mo><mfrac><mn>180</mn><pi/></mfrac></mrow></mfenced></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>A</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>mostrar_etiqueta</mi><mo>=</mo><mi>cert</mi><mo>,</mo><mi>midaPunt</mi><mo>=</mo><mn>7</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>B</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>mostrar_etiqueta</mi><mo>=</mo><mi>cert</mi><mo>,</mo><mi>midaPunt</mi><mo>=</mo><mn>7</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo>-</mo><mn>8</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AB</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AD</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mo>-</mo><mn>5</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1.6815</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>96</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>84</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi>A</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>B</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="inputCompound">true</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> La situació és #G2.

El vèrtex D és l'extrem del vector CD que té per origen C i que és equipol·lent al vector BA.

]]> L'angle A, en un paral·lelogram és 180º - B

]]> 1MA.06.2.14Q 4t vèrtex, costats i angles paral·lelogram Els 3 vèrtex del paral·lelogram ABCD són «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»C«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math».

Determina el quart vèrtex D, quant mesuren els costats CD i BC i quant mesuren els angles A i B

Format:

D=(1,2)

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«mi»D«/mi»«mo»=«/mo»«msqrt»«mn»6«/mn»«/msqrt»«mspace linebreak=¨newline¨/»«mi»B«/mi»«mi»C«/mi»«mo»=«/mo»«msqrt»«mn»11«/mn»«/msqrt»«/math»

A= arrodonit en graus: 25º

B = arrodonit en graus: 25º

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align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mo>-</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c1</mi><mo>,</mo><mi>c2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>v1</mi><mo>=</mo><mi>b1</mi><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>v2</mi><mo>=</mo><mi>b2</mi><mo>-</mo><mi>a2</mi></mtd></mtr><mtr><mtd><mi>u1</mi><mo>=</mo><mi>c1</mi><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>u2</mi><mo>=</mo><mi>c2</mi><mo>-</mo><mi>a2</mi></mtd></mtr><mtr><mtd><mi>r2</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>A</mi><mo>,</mo><mfenced close="]" open="["><mrow><mi>v1</mi><mo>,</mo><mi>v2</mi></mrow></mfenced><mo>)</mo></mtd></mtr></mtable><mrow><mfenced><mrow><mi>v1</mi><mo>*</mo><mi>u1</mi><mo>≠</mo><mn>0</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>distància</mi><mfenced><mrow><mi>C</mi><mo>,</mo><mi>r2</mi></mrow></mfenced><mo>></mo><mn>3</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>u1</mi><mo>*</mo><mi>v2</mi><mo>≠</mo><mi>u2</mi><mo>*</mo><mi>v1</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>c1</mi><mo>-</mo><mi>v1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi><mo>=</mo><mi>c2</mi><mo>-</mo><mi>v2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>punt</mi><mfenced><mrow><mi>d1</mi><mo>,</mo><mi>d2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AB</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>v1</mi><mo>,</mo><mi>v2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AD</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>d1</mi><mo>-</mo><mi>a1</mi><mo>,</mo><mi>d2</mi><mo>-</mo><mi>a2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BA1</mi><mo>=</mo><mi>a1</mi><mo>-</mo><mi>b1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BA2</mi><mo>=</mo><mi>a2</mi><mo>-</mo><mi>b2</mi></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>31</mn><mo> </mo><csymbol definitionURL="http://.../units/degree/angular">°</csymbol></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>9</mn><mo>,</mo><mo>-</mo><mn>7</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msqrt><mn>5</mn></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol5</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>89</mn></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi>C</mi><mi>D</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn><mspace linebreak="newline"/><mi>B</mi><mi>C</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>5</mn><mspace linebreak="newline"/><mi>A</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>B</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="inputCompound">true</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> La situació és #G2.

El vèrtex D és l'extrem del vector CD que té per origen C i que és equipol·lent al vector BA.

Per determinar la longitud dels costats, només cal calcular el mòdul del vector corresponent.

]]> L'angle A, en un paral·lelogram és 180º - B

]]> 1MA.06.2.15Q 4t vèrtex i àrea paral·lelogram Els 3 vèrtex del paral·lelogram ABCD són «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»C«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math».

Determina el quart vèrtex D i quant mesura la seva àrea S (és base·altura)

Format:

D=(1,2)

S= arrodonida

]]> 1.0000000 0.5000000 0 0 D=#sol1S=#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mi style="color:#ffc800">variables</mi><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mo>-</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c1</mi><mo>,</mo><mi>c2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>v1</mi><mo>=</mo><mi>b1</mi><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>v2</mi><mo>=</mo><mi>b2</mi><mo>-</mo><mi>a2</mi></mtd></mtr><mtr><mtd><mi>u1</mi><mo>=</mo><mi>c1</mi><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>u2</mi><mo>=</mo><mi>c2</mi><mo>-</mo><mi>a2</mi></mtd></mtr><mtr><mtd><mi>r2</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>A</mi><mo>,</mo><mfenced close="]" open="["><mrow><mi>v1</mi><mo>,</mo><mi>v2</mi></mrow></mfenced><mo>)</mo></mtd></mtr></mtable><mrow><mfenced><mrow><mi>v1</mi><mo>*</mo><mi>u1</mi><mo>≠</mo><mn>0</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>distància</mi><mfenced><mrow><mi>C</mi><mo>,</mo><mi>r2</mi></mrow></mfenced><mo>></mo><mn>3</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>u1</mi><mo>*</mo><mi>v2</mi><mo>≠</mo><mi>u2</mi><mo>*</mo><mi>v1</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>c1</mi><mo>-</mo><mi>v1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi><mo>=</mo><mi>c2</mi><mo>-</mo><mi>v2</mi></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>C</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blaul</mi><mo>,</mo><mi>mostrar_etiqueta</mi><mo>=</mo><mi>cert</mi><mo>,</mo><mi>midaPunt</mi><mo>=</mo><mn>7</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>D</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>mostrar_etiqueta</mi><mo>=</mo><mi>cert</mi><mo>,</mo><mi>midaPunt</mi><mo>=</mo><mn>7</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>Q1</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>midaPunt</mi><mo>=</mo><mn>9</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>segment</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>AB</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mo>*</mo><mi>x</mi><mo>-</mo><mfrac><mn>7</mn><mn>3</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AD</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AD</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A1</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>8</mn><mo>,</mo><mo>-</mo><mn>8</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>23</mn><mo>*</mo><msqrt><mn>34</mn></msqrt></mrow><mn>34</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>34</mn></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>23</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mfrac><mn>157</mn><mn>34</mn></mfrac><mo>,</mo><mo>-</mo><mfrac><mn>341</mn><mn>34</mn></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>157</mn><mn>34</mn></mfrac><mo>,</mo><mo>-</mo><mfrac><mn>341</mn><mn>34</mn></mfrac></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>S</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="inputCompound">true</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> La situació és #G2.

El vèrtex D és l'extrem del vector CD que té per origen C i que és equipol·lent al vector BA.

Si la base és AB, l'altura és la distància (en vermell) de D a la recta AB (color cian) i es calcula amb la fórmula de la distància d'un punt a un pla.

]]> 1MA.06.2.20DT SIMETRIA RESPECTE PUNT

Simetria respecte a un punt

Si S és el simètric de A respecte a B, S és l'extrem d'un vector doble del vector AB, i les coordenades de S es calculen amb :

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced mathcolor=¨#003300¨»«mrow»«msub»«mi mathvariant=¨bold¨»x«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mo mathvariant=¨bold¨»,«/mo»«msub»«mi mathvariant=¨bold¨»y«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#183;«/mo»«mfenced mathcolor=¨#003300¨»«mrow»«msub»«mi mathvariant=¨bold¨»x«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mo mathvariant=¨bold¨»,«/mo»«msub»«mi mathvariant=¨bold¨»y«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»y«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/mrow»«/mfenced»«/math»

]]> 0.0000000 0.0000000 0 1MA.06.2.21Q Simètric respecte a un punt Troba el punt simètric del punt «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mfenced mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math» respecte al punt «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mfenced mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math»:

#G1

]]> #G2

]]> 1MA.06.2.30DT SIMETRIA RESPECTE A RECTA

Punt simètric respecte a una recta

P és el peu de la projecció ortogonal de A sobre la recta

La projecció ortogonal, és el peu de la perpendicular traçada des del punt A fins a la recta r. És un punt. Format (2,3)

]]> Gràficament: #G1

Primer es calcula l'equació de la perpendicular (blava) a la recta r (negra) que passa per A i que té per vector director «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»vp«/mi»«/mrow»«/mstyle»«/math» (perpendicular a r).

Quan es té l'equació de la perpendicular, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»r«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»2«/mn»«/mrow»«/mstyle»«/math» , n'hi ha prou amb resoldre el sistema d'equacions de les dues rectes per a trobar el punt O

]]> 1MA.06.2.32Q SimètricRecta Determina les coordenades del punt simètric del punt «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mfenced mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math» respecte a la recta «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»r«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8801;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»e«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mstyle»«/math»

Si O és la projecció ortogonal de A sobre r, el simètric de A respecte a r és el simètric de A respecte a O. És un punt. Format (2,3)

]]> Gràficament: #G1

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="en">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mfenced><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mfenced><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>e11</mi><mo>=</mo><mi>aleatori</mi><mfenced><mrow><mn>1</mn><mo>,</mo><mn>5</mn></mrow></mfenced><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>e12</mi><mo>=</mo><mi>aleatori</mi><mfenced><mrow><mn>1</mn><mo>,</mo><mn>5</mn></mrow></mfenced><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>e13</mi><mo>=</mo><mi>aleatori</mi><mfenced><mrow><mn>1</mn><mo>,</mo><mn>5</mn></mrow></mfenced><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>e1</mi><mo>=</mo><mi>e11</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>e12</mi><mo>*</mo><mi>y</mi><mo>+</mo><mi>e13</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>r</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>e1</mi><mo>)</mo></mtd></mtr></mtable><mrow><mi>distància</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>⩾</mo><mn>2</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>vp</mi><mo>=</mo><mfenced close="]" open="["><mrow><mo>-</mo><mi>e11</mi><mo>,</mo><mo>-</mo><mi>e12</mi></mrow></mfenced></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>2</mn><mo>*</mo><msub><mi>T</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>-</mo><mi>a1</mi><mo>,</mo><mn>2</mn><mo>*</mo><msub><mi>T</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>-</mo><mi>a2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>S</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>A</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>mostrar_etiqueta</mi><mo>=</mo><mi>cert</mi></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>O</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>mostrar_etiqueta</mi><mo>=</mo><mi>cert</mi><mo>,</mo><mi>color</mi><mo>=</mo><mi>vermell</mi></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>S</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>mostrar_etiqueta</mi><mo>=</mo><mi>cert</mi><mo>,</mo><mi>mida_punt</mi><mo>=</mo><mn>7</mn><mo>,</mo><mi>color</mi><mo>=</mo><mi>vermell</mi></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>r</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>negre</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>r2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mo>*</mo><mi>x</mi><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>5</mn></mfrac><mo>*</mo><mi>x</mi><mo>+</mo><mfrac><mn>28</mn><mn>5</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>87</mn><mn>17</mn></mfrac><mo>,</mo><mfrac><mn>43</mn><mn>17</mn></mfrac></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>52</mn><mn>17</mn></mfrac><mo>,</mo><mfrac><mn>64</mn><mn>17</mn></mfrac></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> La situació és #G1

El simètric S és l'extrem d'un vector AS amb origen a A i de components dobles de les components de AO.

]]> 1MA.06.2.40DT ROMBE

Rombe

Té els 4 costats iguals i els angles oposats iguals. Dos angles adjacents sumen 180º.

Perímetre:P= 4c.

Àrea: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold-italic¨ mathcolor=¨#7F007F¨»S«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#7F007F¨»=«/mo»«mfrac mathcolor=¨#7F007F¨»«mrow»«mi mathvariant=¨bold¨»D«/mi»«mi mathvariant=¨bold¨»d«/mi»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math»

Les diagonals D i d són perpendiculars i es tallen en el seu punt mitjà.

]]> 0.0000000 0.0000000 0 1MA 06. LLOCS GEOMÈTRICS/1MA.06.3 Còniques 1MA.06.3.10DT EQUACIÓ DE LA CIRCUMFERÈNCIA Equació de la circumferència

L'equació d'una circumferència de centre C(a,b) i de radi r es calcula amb:

x² + y² - 2ax - 2by + a² + b² - r²= 0

A partir de l'equació de la circumferència podem deduir el centre (a,b) i el radi r,
trobant "a" a partir de -2ax
trobant "b" a partir de -2by
i deduint "r" a partir de a² + b² -r² ]]> 0.0000000 0.0000000 0 1MA.06.3.11Q EqCircumferència Troba l'equació de la circumferència de radi #r i de centre C(#a,#b)

Format de la resposta: x²+y²-4x-y+15=0

]]> #G1

el coeficient de y és -2·#b

el terme independent es calcula amb (#a)² + (#b)² - (#r)²

]]> 1MA.06.3.21Q TrobarCentreRadi Troba les coordenades del centre i el radi de la circumferència d'equació

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»e_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mrow»«/mstyle»«/math»

]]> 1.0000000 0.3333333 0 0 a) centre =#sol1b) radi =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_11</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_11</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_11</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi><mo>=</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_11</mi><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mi>b_11</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mi>r_11</mi></mfenced><mn>2</mn></msup><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a_11</mi><mo>,</mo><mi>b_11</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>r_11</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w1</mi><mo>=</mo><mo> </mo><mo>(</mo><mi>a_11</mi><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mi>b_11</mi><msup><mo>)</mo><mn>2</mn></msup><mo>-</mo><msup><mfenced><mi>r_11</mi></mfenced><mn>2</mn></msup></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mo>*</mo><mi>x</mi><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><mi>y</mi><mo>+</mo><mn>36</mn><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>6</mn><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>36</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>c</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>r</mi><mi>a</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Com que l'equació oficial de la recta és:

x² + y² - 2ax - 2by + a² + b² - r² = 0

identifica a i b amb els coeficients de x i de y.

Després, calcula r ja que a² + b² - r² = #w1

]]> 1MA.06.3.31Q RectaTangentPuntCircumf Considera la circumferència de centre C (#a,#b) i de radi #r.

Troba l'equació explícita de la recta tangent en el punt P (#p,#q).
Pensa que si la recta és tangent, és perpendicular al vector radi «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover accent=¨true¨»«mrow»«mi»C«/mi»«mi»P«/mi»«/mrow»«mo»§#8594;«/mo»«/mover»«/math»

]]> #G1

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol 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align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>P</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>mida_punt</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>A1</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>mida_punt</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>segment</mi><mo>(</mo><mi>P</mi><mo>,</mo><mi>A1</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>recta</mi><mfenced><mrow><mi>P</mi><mo>,</mo><mi>pp</mi></mrow></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>20</mn><mo>,</mo><mn>2</mn><mo>*</mo><msqrt><mn>5</mn></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>12</mn><mo>*</mo><mi>y</mi><mo>+</mo><mn>40</mn><mo>=</mo><mn>20</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><mn>8</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mn>4</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mn>4</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_62</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_62</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> És una recta que passa per P i que és perpendicular al vector «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mover»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»CP«/mi»«mo»§#8594;«/mo»«/mover»«/mstyle»«/math»

]]> 1MA.06.3.32Q TangentsDesDePunt Troba l'equació de les rectes tangents a la circumferència de radi #r i de centre C(#a,#b), traçades dels del punt P(#p_1,#p_2).

Primer cal escriure l'equació d'una recta que passi per p (y=mx+n, on el punt ens permet trobar n en funció de m).
Després es resol el sistema format per les dues equacions per substitució; queda una equació de 2n grau. Cal pensar que si són tangents, el punt de contacte és únic, i per tant l'equació de 2n grau té solució doble; això ens permet calcular m.

Format de la resposta: {y=2x+4,y=4x-9}

]]> #G1

L'equació es dedueix del fet que un punt qualsevol de la paràbola és equidistant del focus F(x_F,y_F) i d'una recta r (anomenada directriu) d'equació ax+by+c=0. I per tant es troba igualant:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfenced mathcolor=¨#003300¨ open=¨|¨ close=¨|¨»«mfenced open=¨|¨ close=¨|¨»«mover»«mi mathvariant=¨bold¨»XF«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mfenced»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»d«/mi»«mfenced mathcolor=¨#003300¨»«mrow»«mi mathvariant=¨bold¨»X«/mi»«mo mathvariant=¨bold¨»,«/mo»«mi mathvariant=¨bold¨»d«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8660;«/mo»«msqrt mathcolor=¨#003300¨»«msup»«mfenced»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mi mathvariant=¨bold¨»F«/mi»«/msub»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«mfenced»«mrow»«mi mathvariant=¨bold¨»y«/mi»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»y«/mi»«mi mathvariant=¨bold¨»F«/mi»«/msub»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«msub»«mi mathvariant=¨bold¨»ax«/mi»«mi mathvariant=¨bold¨»F«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»by«/mi»«mi mathvariant=¨bold¨»F«/mi»«/msub»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»c«/mi»«/mrow»«/mfenced»«msqrt»«msup»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mfrac»«/mrow»«/mstyle»«/math»

]]> 0.0000000 0.0000000 0 1MA.06.3.50DT EQUACIÓ DE L'EL·LIPSE Equació de l'el·lipse

L'equació es dedueix del fet que un punt qualsevol de l'el·lipse és tal que la suma de les seves distàncies als dos focus és constant. Si l'eix major és 2a i l'eix menor és 2b, l'equació d'una el·lipse centrada a l'origen és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#003300¨»«msup»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«msup»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mfrac mathcolor=¨#003300¨»«msup»«mi mathvariant=¨bold¨»y«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«msup»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mrow»«/mstyle»«/math»

Per dibuixar-la, és interessant veure que es pot fer de moltes maneres a la pàgina de Pere Planells

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 0.0000000 0.0000000 0 1MA1.06.3.41Q EqParàbola Troba l'equació de la paràbola de focus situat en el punt «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»0«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math» i de directriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»n«/mi»«/mrow»«/mstyle»«/math»

Format de la resposta: y²=4x+15

]]> El gràfic és #G

La distància d'un punt qualsevol P(x,y) a la recta d'equació «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«/mrow»«/mstyle»«/math»" src="http://www.insmilaifontanals.cat/moodle/lib/editor/tinymce/plugins/tiny_mce_wiris/tinymce/integration/showimage.php?formula=d8b7c64b9b656d9760172084baaf3535&cw=72&ch=10&cb=9" style="vertical-align: -1px;"/> es calcula amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfrac mathcolor=¨#0000FF¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»m«/mi»«/mrow»«/mfenced»«msqrt»«mn mathvariant=¨bold¨»1«/mn»«/msqrt»«/mfrac»«/mstyle»«/math»" src="http://www.insmilaifontanals.cat/moodle/lib/editor/tinymce/plugins/tiny_mce_wiris/tinymce/integration/showimage.php?formula=d9f2d440459548d4b453aa6281776b84&cw=47&ch=32&cb=18" style="vertical-align: -14px;"/>

S'igualen les dues expressions i s'eleva al quadrat per treure l'arrel.

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