1MA 05. RECTES EN EL PLA/1MA.05.4 Angles i distàncies 1MA.05.4.10DT Angle entre rectes

Angle entre rectes

L'angle entre dues rectes és el que formen els seus vectors directors (o el seu suplementari, si l'angle és obtús)

L'angle entre dos vectors és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfenced mathcolor=¨#003300¨»«mover»«mrow»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#8640;«/mo»«/mover»«mo mathvariant=¨bold¨»,«/mo»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#8640;«/mo»«/mover»«/mrow»«mo mathvariant=¨bold¨»^«/mo»«/mover»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»acos«/mi»«mfrac mathcolor=¨#003300¨»«mrow»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#8640;«/mo»«/mover»«mo mathvariant=¨bold¨»§#183;«/mo»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#8640;«/mo»«/mover»«/mrow»«mrow»«mfenced open=¨|¨ close=¨|¨»«mfenced open=¨|¨ close=¨|¨»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#8640;«/mo»«/mover»«/mfenced»«/mfenced»«mo mathvariant=¨bold¨»§#183;«/mo»«mfenced open=¨|¨ close=¨|¨»«mfenced open=¨|¨ close=¨|¨»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#8640;«/mo»«/mover»«/mfenced»«/mfenced»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»

]]> 0.0000000 0.0000000 0 1MA.05.4.11Q AngleRectes Cont_Cont Considera les dues rectes?
r: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»12«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» i s: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»21«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»22«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math».

a) Troba el vector de r inclòs en l'enunciat

b) Troba el vector de s inclòs en l'enunciat

c) Determina l'angle entre les dues rectes
Format de la resposta:

vector: [1,2]

angle en graus, arrodonit a la unitat:37º

]]> Gràficament: #G1

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xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>vr</mi><mo>,</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><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><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>vs</mi><mo>,</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t1</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>e11</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>7</mn><mo>*</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e12</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>6</mn><mo>*</mo><mi>y</mi><mo>=</mo><mn>70</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>7</mn><mo>,</mo><mn>1</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>vs</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mn>6</mn><mo>,</mo><mn>2</mn></mrow></mfenced></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"><msub><mi>v</mi><mi>r</mi></msub><mo> </mo><mo>=</mo><mo>#</mo><mi>v</mi><mi>r</mi><mspace linebreak="newline"/><msub><mi>v</mi><mi>s</mi></msub><mo> </mo><mo>=</mo><mo>#</mo><mi>v</mi><mi>s</mi><mspace linebreak="newline"/><mi>a</mi><mi>n</mi><mi>g</mi><mi>l</mi><mi mathvariant="normal">e</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</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> Un vector de r és #vr

Un vector de s és #vs

L'angle es calcula amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»acos«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mfrac mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mrow»«msqrt»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»2«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«msqrt»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»acos«/mi»«mfrac mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»pe«/mi»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»pm«/mi»«/mrow»«/mfrac»«/mstyle»«/math»

]]> 1MA.05.4.12Q AngleRectes Cont_Cart Considera les dues rectes
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»r«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8801;«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»12«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/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»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»s«/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¨»2«/mn»«/mrow»«/mstyle»«/math»

a) Troba el vector de r inclòs en l'enunciat

b) Troba el vector de s inclòs en l'enunciat

c) Determina l'angle entre les dues rectes

Format de la resposta:

vectors: [1,2]

angle en graus, arrodonit a la unitat: 54º

]]> Gràficament: #G1

]]> 1.0000000 0.5000000 0 0 vr =#vrvs =#vsangle =#sol4]]> <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>t1</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>t1</mi><mo>=</mo><mn>1</mn></mrow><mtable><mtr><mtd><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>u1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</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>u2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</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><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>b2</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>v1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</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>v2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</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>e11</mi><mo>=</mo><mi>u2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>u1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>a1</mi><mo>*</mo><mi>u2</mi><mo>-</mo><mi>a2</mi><mo>*</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>e12</mi><mo>=</mo><mi>v2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>v1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>b1</mi><mo>*</mo><mi>v2</mi><mo>-</mo><mi>b2</mi><mo>*</mo><mi>v1</mi></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>rang</mi><mfenced><mtable><mtr><mtd><mi>u2</mi></mtd><mtd><mo>-</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>v2</mi></mtd><mtd><mo>-</mo><mi>v1</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>2</mn></mrow></apply></mtd></mtr><mtr><mtd/></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mo>"</mo><mi>Els</mi><mo> </mo><mi>vectors</mi><mo> </mo><mi>són</mi><mo> </mo><mi>independents</mi><mo>"</mo></mtd></mtr><mtr><mtd/></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>t1</mi><mo>=</mo><mn>2</mn></mrow><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</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>9</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>u1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</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>u2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</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>a1</mi></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>a2</mi></mtd></mtr><mtr><mtd><mi>m1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</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>v1</mi><mo>=</mo><mi>m1</mi><mo>*</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>v2</mi><mo>=</mo><mi>m1</mi><mo>*</mo><mi>u2</mi></mtd></mtr><mtr><mtd><mi>e11</mi><mo>=</mo><mi>u2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>u1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>a1</mi><mo>*</mo><mi>u2</mi><mo>-</mo><mi>a2</mi><mo>*</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>e12</mi><mo>=</mo><mi>v2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>v1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>b1</mi><mo>*</mo><mi>v2</mi><mo>-</mo><mi>b2</mi><mo>*</mo><mi>v1</mi></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mo>"</mo><mi>Els</mi><mo> </mo><mi>vectors</mi><mo> </mo><mi>són</mi><mo> </mo><mi>dependents</mi><mo>;</mo><mo> </mo><mi>cal</mi><mo> </mo><mi>esbrinar</mi><mo> </mo><mi>si</mi><mo> </mo><mi>un</mi><mo> </mo><mi>punt</mi><mo> </mo><mi>de</mi><mo> </mo><mi>r</mi><mo> </mo><mi>pertany</mi><mo> </mo><mi>a</mi><mo> </mo><mi>s</mi><mo>"</mo></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mtable><mtr><mtd><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>u1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</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>u2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</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>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>b2</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>m1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</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>v1</mi><mo>=</mo><mi>m1</mi><mo>*</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>v2</mi><mo>=</mo><mi>m1</mi><mo>*</mo><mi>u2</mi></mtd></mtr><mtr><mtd><mi>e11</mi><mo>=</mo><mi>u2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>u1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>a1</mi><mo>*</mo><mi>u2</mi><mo>-</mo><mi>a2</mi><mo>*</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>e12</mi><mo>=</mo><mi>v2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>v1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>b1</mi><mo>*</mo><mi>v2</mi><mo>-</mo><mi>b2</mi><mo>*</mo><mi>v1</mi></mtd></mtr><mtr><mtd/></mtr></mtable><mfenced><mrow><mfrac><mrow><mi>a1</mi><mo>*</mo><mi>u2</mi><mo>-</mo><mi>a2</mi><mo>*</mo><mi>u1</mi></mrow><mrow><mi>b1</mi><mo>*</mo><mi>v2</mi><mo>-</mo><mi>b2</mi><mo>*</mo><mi>v1</mi></mrow></mfrac><mo>≠</mo><mfrac><mi>u2</mi><mi>v2</mi></mfrac></mrow></mfenced></apply></mtd></mtr><mtr><mtd/></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mo>"</mo><mi>Els</mi><mo> </mo><mi>vectors</mi><mo> </mo><mi>són</mi><mo> </mo><mi>dependents</mi><mo>;</mo><mo> </mo><mi>cal</mi><mo> </mo><mi>esbrinar</mi><mo> </mo><mi>si</mi><mo> </mo><mi>un</mi><mo> </mo><mi>punt</mi><mo> </mo><mi>de</mi><mo> </mo><mi>r</mi><mo> </mo><mi>pertany</mi><mo> </mo><mi>a</mi><mo> </mo><mi>s</mi><mo>"</mo></mtd></mtr></mtable></apply></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>vr</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>u1</mi><mo>,</mo><mi>u2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>vs</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>n11</mi><mo>=</mo><mi>x</mi><mo>-</mo><mi>a1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n12</mi><mo>=</mo><mi>y</mi><mo>-</mo><mi>a2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n21</mi><mo>=</mo><mi>x</mi><mo>-</mo><mi>b1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n22</mi><mo>=</mo><mi>y</mi><mo>-</mo><mi>b2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e2</mi><mo>=</mo><mi>e12</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pe</mi><mo>=</mo><apply><scalarproduct/><mi>vr</mi><mi>vs</mi></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pm</mi><mo>=</mo><mfenced close="‖" open="‖"><mi>vr</mi></mfenced><mo>*</mo><mfenced close="‖" open="‖"><mi>vs</mi></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>co</mi><mo>=</mo><mfrac><mi>pe</mi><mi>pm</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfenced><mrow><mi>acos</mi><mfrac><mi>pe</mi><mi>pm</mi></mfrac></mrow></mfenced><mo>*</mo><mfrac><mn>180</mn><pi/></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>arrodoneix</mi><mo>(</mo><mi>sol</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>sol2</mi><mo>⩽</mo><mn>90</mn></mrow><mrow><mi>sol3</mi><mo>=</mo><mi>sol2</mi></mrow><mrow><mi>sol3</mi><mo>=</mo><mn>180</mn><mo>-</mo><mi>sol2</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi><mo>=</mo><mi>sol3</mi><csymbol definitionURL="http://.../units/degree/angular">°</csymbol></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mo>(</mo><mfenced close="}" open="{"><mtable 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align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</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>vr</mi><mo>,</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><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><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>vs</mi><mo>,</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t1</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"><mo>(</mo><mi>u1</mi><mo>,</mo><mi>u2</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>,</mo><mo>-</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>v1</mi><mo>,</mo><mi>v2</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>,</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n11</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n12</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>+</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e12</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>*</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>19</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><mn>4</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>vs</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mn>2</mn><mo>,</mo><mn>3</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>72</mn><mo> </mo><csymbol definitionURL="http://.../units/degree/angular">°</csymbol></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"><msub><mi>v</mi><mi>r</mi></msub><mo> </mo><mo>=</mo><mo>#</mo><mi>v</mi><mi>r</mi><mspace linebreak="newline"/><msub><mi>v</mi><mi>s</mi></msub><mo> </mo><mo>=</mo><mo>#</mo><mi>v</mi><mi>s</mi><mspace linebreak="newline"/><mi>a</mi><mi>n</mi><mi>g</mi><mi>l</mi><mi mathvariant="normal">e</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</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> Un vector de r és #vr

Un vector de s és #vs

L'angle es calcula amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#00066F¨»acos«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00066F¨»§#160;«/mo»«mfrac mathcolor=¨#00066F¨»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mrow»«msqrt»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»2«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«mo»§#160;«/mo»«mo»§#160;«/mo»«msqrt»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#00066F¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00066F¨»acos«/mi»«mfrac mathcolor=¨#00066F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»pe«/mi»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»pm«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»

La situació és aquesta: #G1

]]> 1MA.05.4.13Q AngleRectes Expl_Cart Quin angle formen les dues rectes?
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mstyle displaystyle=¨false¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»r«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«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»«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»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»s«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«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¨»2«/mn»«/mstyle»«/mstyle»«/math»

Format de la resposta: angle en graus, arrodonit a la unitat: 34º

]]> Gràficament: #G1

]]> 1.0000000 0.5000000 0 0 #sol4 <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>t1</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>t1</mi><mo>=</mo><mn>1</mn></mrow><mtable><mtr><mtd><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>9</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>9</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>u1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</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>u2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</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>9</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>9</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>v1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</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>v2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</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>e11</mi><mo>=</mo><mi>u2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>u1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>a1</mi><mo>*</mo><mi>u2</mi><mo>-</mo><mi>a2</mi><mo>*</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>e12</mi><mo>=</mo><mi>v2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>v1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>b1</mi><mo>*</mo><mi>v2</mi><mo>-</mo><mi>b2</mi><mo>*</mo><mi>v1</mi></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>rang</mi><mfenced><mtable><mtr><mtd><mi>u2</mi></mtd><mtd><mo>-</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>v2</mi></mtd><mtd><mo>-</mo><mi>v1</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>2</mn></mrow></apply></mtd></mtr><mtr><mtd/></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mo>"</mo><mi>Els</mi><mo> </mo><mi>pendents</mi><mo> </mo><mi>són</mi><mo> </mo><mi>diferents</mi><mo>"</mo></mtd></mtr><mtr><mtd/></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>t1</mi><mo>=</mo><mn>2</mn></mrow><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</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>9</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>u1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</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>u2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</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>a1</mi></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>a2</mi></mtd></mtr><mtr><mtd><mi>m1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</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>v1</mi><mo>=</mo><mi>m1</mi><mo>*</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>v2</mi><mo>=</mo><mi>m1</mi><mo>*</mo><mi>u2</mi></mtd></mtr><mtr><mtd><mi>e11</mi><mo>=</mo><mi>u2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>u1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>a1</mi><mo>*</mo><mi>u2</mi><mo>-</mo><mi>a2</mi><mo>*</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>e12</mi><mo>=</mo><mi>v2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>v1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>b1</mi><mo>*</mo><mi>v2</mi><mo>-</mo><mi>b2</mi><mo>*</mo><mi>v1</mi></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mo>"</mo><mi>Els</mi><mo> </mo><mi>pendents</mi><mo> </mo><mi>són</mi><mo> </mo><mi>iguals</mi><mo>;</mo><mo> </mo><mi>cal</mi><mo> </mo><mi>esbrinar</mi><mo> </mo><mi>si</mi><mo> </mo><mi>un</mi><mo> </mo><mi>punt</mi><mo> </mo><mi>de</mi><mo> </mo><mi>r</mi><mo> </mo><mi>pertany</mi><mo> </mo><mi>a</mi><mo> </mo><mi>s</mi><mo>"</mo></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mtable><mtr><mtd><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>9</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>9</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>u1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</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>u2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</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>9</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>9</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>m1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</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>v1</mi><mo>=</mo><mi>m1</mi><mo>*</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>v2</mi><mo>=</mo><mi>m1</mi><mo>*</mo><mi>u2</mi></mtd></mtr><mtr><mtd><mi>e11</mi><mo>=</mo><mi>u2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>u1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>a1</mi><mo>*</mo><mi>u2</mi><mo>-</mo><mi>a2</mi><mo>*</mo><mi>u1</mi></mtd></mtr><mtr><mtd><mi>e12</mi><mo>=</mo><mi>v2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>v1</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>b1</mi><mo>*</mo><mi>v2</mi><mo>-</mo><mi>b2</mi><mo>*</mo><mi>v1</mi></mtd></mtr><mtr><mtd/></mtr></mtable><mfenced><mrow><mfrac><mrow><mi>a1</mi><mo>*</mo><mi>u2</mi><mo>-</mo><mi>a2</mi><mo>*</mo><mi>u1</mi></mrow><mrow><mi>b1</mi><mo>*</mo><mi>v2</mi><mo>-</mo><mi>b2</mi><mo>*</mo><mi>v1</mi></mrow></mfrac><mo>≠</mo><mfrac><mi>u2</mi><mi>v2</mi></mfrac></mrow></mfenced></apply></mtd></mtr><mtr><mtd/></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mo>"</mo><mi>Els</mi><mo> </mo><mi>pendents</mi><mo> </mo><mi>són</mi><mo> </mo><mi>iguals</mi><mo>;</mo><mo> </mo><mi>cal</mi><mo> </mo><mi>esbrinar</mi><mo> </mo><mi>si</mi><mo> </mo><mi>un</mi><mo> </mo><mi>punt</mi><mo> </mo><mi>de</mi><mo> </mo><mi>r</mi><mo> </mo><mi>pertany</mi><mo> </mo><mi>a</mi><mo> </mo><mi>s</mi><mo>"</mo></mtd></mtr></mtable></apply></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pr</mi><mo>=</mo><mfrac><mi>u2</mi><mi>u1</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ps</mi><mo>=</mo><mfrac><mi>v2</mi><mi>v1</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e1</mi><mo>=</mo><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mi>u2</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a1</mi><mo>*</mo><mi>u2</mi><mo>-</mo><mi>a2</mi><mo>*</mo><mi>u1</mi></mrow><mrow><mo>-</mo><mi>u1</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e2</mi><mo>=</mo><mi>v2</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>v1</mi><mo>*</mo><mi>y</mi><mo>-</mo><mi>b1</mi><mo>*</mo><mi>v2</mi><mo>+</mo><mi>b2</mi><mo>*</mo><mi>v1</mi><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e3</mi><mo>=</mo><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mi>v2</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>b1</mi><mo>*</mo><mi>v2</mi><mo>-</mo><mi>b2</mi><mo>*</mo><mi>v1</mi></mrow><mrow><mo>-</mo><mi>v1</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>vr</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>u1</mi><mo>,</mo><mi>u2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>vs</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>pe</mi><mo>=</mo><apply><scalarproduct/><mi>vr</mi><mi>vs</mi></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pm</mi><mo>=</mo><mfenced close="‖" open="‖"><mi>vr</mi></mfenced><mo>*</mo><mfenced close="‖" open="‖"><mi>vs</mi></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>co</mi><mo>=</mo><mfrac><mi>pe</mi><mi>pm</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfenced><mrow><mi>acos</mi><mfrac><mi>pe</mi><mi>pm</mi></mfrac></mrow></mfenced><mo>*</mo><mfrac><mn>180</mn><pi/></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>arrodoneix</mi><mo>(</mo><mi>sol</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>sol2</mi><mo>⩽</mo><mn>90</mn></mrow><mrow><mi>sol3</mi><mo>=</mo><mi>sol2</mi></mrow><mrow><mi>sol3</mi><mo>=</mo><mn>180</mn><mo>-</mo><mi>sol2</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi><mo>=</mo><mi>sol3</mi><csymbol definitionURL="http://.../units/degree/angular">°</csymbol></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>amplada</mi><mo>=</mo><mn>20</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>20</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>recta</mi><mo>(</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo><mo>,</mo><mi>vr</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>1</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>recta</mi><mo>(</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo><mo>,</mo><mi>vs</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</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>vr</mi><mo>,</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><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><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>vs</mi><mo>,</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t1</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"><mo>(</mo><mi>u1</mi><mo>,</mo><mi>u2</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>,</mo><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>v1</mi><mo>,</mo><mi>v2</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo>-</mo><mn>9</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>7</mn><mn>5</mn></mfrac><mo>*</mo><mi>x</mi><mo>-</mo><mfrac><mn>73</mn><mn>5</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>9</mn><mo>*</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mn>68</mn><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>9</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>68</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pr</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>5</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ps</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>9</mn></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"><mn>42</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol4</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> Un vector de r és #vr

Un vector de s és #vs

L'angle es calcula amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»acos«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mfrac mathcolor=¨#000066¨»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mrow»«msqrt»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»2«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«mo mathvariant=¨bold¨»§#183;«/mo»«msqrt»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»acos«/mi»«mfrac mathcolor=¨#000066¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»pe«/mi»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»pm«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»

La situació és aquesta: #G1

]]> 1MA.05.4.14Q AngleRectes Para_Cont Quin angle formen les rectes «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»r«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8801;«/mo»«mfenced mathcolor=¨#003300¨ open=¨{¨ close=¨}¨»«mtable»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»12«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«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¨»s«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8801;«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»21«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»22«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» ?

Format de la resposta:

angle en graus arrodonit a la unitat: 34º

]]> Gràficament: #G1

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align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</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>vr</mi><mo>,</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><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><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>vs</mi><mo>,</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t1</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>e11</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>*</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>22</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e12</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>8</mn><mo>*</mo><mi>y</mi><mo>=</mo><mn>41</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><mn>3</mn><mo>,</mo><mn>4</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>vs</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>9</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"><mn>78.503</mn></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"><mn>79</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pe</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pm</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>*</mo><msqrt><mn>145</mn></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>co</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>12</mn><mo>*</mo><msqrt><mn>145</mn></msqrt></mrow><mn>725</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol4</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> Un vector de r és #vr

Un vector de s és #vs

L'angle es calcula amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»acos«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mfrac mathcolor=¨#000066¨»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mrow»«msqrt»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u«/mi»«mn mathvariant=¨bold¨»2«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«mo mathvariant=¨bold¨»§#183;«/mo»«msqrt»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»acos«/mi»«mfrac mathcolor=¨#000066¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»pe«/mi»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»pm«/mi»«/mrow»«/mfrac»«/mstyle»«/math»

La situació és aquesta: #G1

]]> 1MA.05.4.15Q Trobar m/ 2RectesFormenAngle60º Per quin(s) valor(s) de m les 2 rectes que tenen les equacions següents formen un angle de 60º?

«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»«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»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»s«/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¨»2«/mn»«/mstyle»«/math»

Format de la resposta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mfrac»«mrow»«mn mathvariant=¨bold¨»3«/mn»«msqrt»«mn mathvariant=¨bold¨»5«/mn»«/msqrt»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mn mathvariant=¨bold¨»3«/mn»«mn mathvariant=¨bold¨»4«/mn»«/mfrac»«mo mathvariant=¨bold¨»,«/mo»«mfrac»«mrow»«mn mathvariant=¨bold¨»3«/mn»«msqrt»«mn mathvariant=¨bold¨»5«/mn»«/msqrt»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mn mathvariant=¨bold¨»3«/mn»«mn mathvariant=¨bold¨»4«/mn»«/mfrac»«/mrow»«/mfenced»«/mstyle»«/math»

]]> 1.0000000 0.3333333 0 0 #sol1 <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 definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a11</mi><mo>=</mo><mi>m</mi></mtd></mtr><mtr><mtd><mi>b11</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>c11</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>v1</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>b11</mi><mo>,</mo><mi>m</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mi>a12</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>b12</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>c12</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>v2</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>b12</mi><mo>,</mo><mi>a12</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mn>3</mn><mo>*</mo><msup><mfenced><mi>a12</mi></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mi>b12</mi></mfenced><mn>2</mn></msup></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mn>8</mn><mo>*</mo><mi>b11</mi><mo>*</mo><mi>b12</mi><mo>*</mo><mi>a12</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mn>3</mn><mo>*</mo><msup><mfenced><mi>b11</mi></mfenced><mn>2</mn></msup><mo>*</mo><msup><mfenced><mi>b12</mi></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mi>b11</mi></mfenced><mn>2</mn></msup><mo>*</mo><msup><mfenced><mi>a12</mi></mfenced><mn>2</mn></msup></mtd></mtr><mtr><mtd><mi>d</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>c</mi></mtd></mtr></mtable><mfenced><mrow><mi>d</mi><mo>></mo><mn>0</mn></mrow></mfenced></apply></mtd></mtr><mtr><mtd/></mtr><mtr><mtd><mi>m1</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mi>b</mi><mo>+</mo><msqrt><mi>d</mi></msqrt></mrow><mrow><mn>2</mn><mo>*</mo><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mi>m2</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mi>b</mi><mo>-</mo><msqrt><mi>d</mi></msqrt></mrow><mrow><mn>2</mn><mo>*</mo><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mrow><mi>m1</mi><mo>*</mo><mi>m2</mi><mo>∈</mo><rationals/></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>e_1</mi><mo>=</mo><mi>m</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>b11</mi><mo>*</mo><mi>y</mi><mo>+</mo><mi>c11</mi><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_2</mi><mo>=</mo><mi>a12</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>b12</mi><mo>*</mo><mi>y</mi><mo>+</mo><mi>c12</mi><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>m1</mi><mo>,</mo><mi>m2</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><apply><scalarproduct/><mi>v1</mi><mi>v2</mi></apply><mrow><mfenced close="‖" open="‖"><mi>v1</mi></mfenced><mo>*</mo><mfenced close="‖" open="‖"><mi>v2</mi></mfenced></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f11</mi><mo>=</mo><mfenced><mrow><mn>2</mn><mo>*</mo><apply><scalarproduct/><mi>v1</mi><mi>v2</mi></apply></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f12</mi><mo>=</mo><msup><mfenced><mrow><mn>2</mn><mo>*</mo><apply><scalarproduct/><mi>v1</mi><mi>v2</mi></apply></mrow></mfenced><mn>2</mn></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f21</mi><mo>=</mo><mfenced><mrow><mfenced close="‖" open="‖"><mi>v1</mi></mfenced><mo>*</mo><mfenced close="‖" open="‖"><mi>v2</mi></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f22</mi><mo>=</mo><msup><mfenced><mrow><mfenced close="‖" open="‖"><mi>v1</mi></mfenced><mo>*</mo><mfenced close="‖" open="‖"><mi>v2</mi></mfenced></mrow></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"><mi>m</mi><mo>*</mo><mi>x</mi><mo>+</mo><mn>5</mn><mo>*</mo><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>,</mo><mn>40</mn><mo>,</mo><mn>50</mn><mo>,</mo><mn>1200</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>*</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>10</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn><mo>*</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>10</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"><mfrac><mrow><mo>-</mo><mi>m</mi><mo>-</mo><mn>5</mn></mrow><mrow><msqrt><mn>2</mn></msqrt><mo>*</mo><msqrt><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn></mrow></msqrt></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f11</mi><mo>,</mo><mi>f12</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>*</mo><mi>m</mi><mo>-</mo><mn>10</mn><mo>,</mo><mn>4</mn><mo>*</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>40</mn><mo>*</mo><mi>m</mi><mo>+</mo><mn>100</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f21</mi><mo>,</mo><mi>f22</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt><mo>*</mo><msqrt><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn></mrow></msqrt><mo>,</mo><mn>2</mn><mo>*</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>50</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 close="}" open="{"><mtable align="center"><mtr><mtd><mn>5</mn><mo>*</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>10</mn><mo>,</mo><mo>-</mo><mn>5</mn><mo>*</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>10</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol1</correctAnswer></correctAnswers><assertions><assertion name="equivalent_symbolic"/><assertion name="syntax_expression"/></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> Els vectors directors de les rectes són «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mover mathcolor=¨#00007F¨»«mrow»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»v«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mover mathcolor=¨#00007F¨»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«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¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»v«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»2«/mn»«/mstyle»«/math»

Per una banda, el cosinus de l'angle que formen les rectes és

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»cosa«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#8201;«/mo»«mfrac mathcolor=¨#00007F¨»«mrow»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨»§#183;«/mo»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mrow»«mrow»«mfenced open=¨|¨ close=¨|¨»«mfenced open=¨|¨ close=¨|¨»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mfenced»«/mfenced»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mfenced open=¨|¨ close=¨|¨»«mfenced open=¨|¨ close=¨|¨»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mfenced»«/mfenced»«/mrow»«/mfrac»«/mstyle»«/math»

per altra banda, el cosinus de 60º és 1/2.

Cal doncs resoldre:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#8201;«/mo»«mfrac mathcolor=¨#00007F¨»«mrow»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨»§#183;«/mo»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mrow»«mrow»«mfenced open=¨|¨ close=¨|¨»«mfenced open=¨|¨ close=¨|¨»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mfenced»«/mfenced»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mfenced open=¨|¨ close=¨|¨»«mfenced open=¨|¨ close=¨|¨»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mfenced»«/mfenced»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mfrac mathcolor=¨#00007F¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«/mstyle»«/math»

]]> Cal resoldre:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»f«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»f«/mi»«mn mathvariant=¨bold¨»21«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mfrac mathcolor=¨#00007F¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#8660;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#183;«/mo»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»f«/mi»«mn mathvariant=¨bold¨»11«/mn»«/mrow»«/mfenced»«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¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»f«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»21«/mn»«/mrow»«/mstyle»«/math»

Si elevem els dos membres al quadrat, obtenim una equació de 2n grau:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»f«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»12«/mn»«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¨»f«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»22«/mn»«/mrow»«/mstyle»«/math»

]]> 1MA.05.4.21Q Trobar 3 angles d'un triangle Dibuixa 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»,
i determina els angles que corresponen a cada vèrtex.

Format: 54º (arrodonit en graus)

]]> #G1

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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>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></mtd></mtr><mtr><mtd><mi>A1</mi><mo>=</mo><mi>arrodonir</mi><mo>(</mo><mfrac><mn>180</mn><pi/></mfrac><mo>*</mo><mi>angle</mi><mo>(</mo><mi>T</mi><mo>,</mo><mn>1</mn><mo>)</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>B1</mi><mo>=</mo><mi>arrodonir</mi><mfenced><mrow><mfrac><mn>180</mn><pi/></mfrac><mo>*</mo><mi>angle</mi><mo>(</mo><mi>T</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mi>C1</mi><mo>=</mo><mi>arrodonir</mi><mfenced><mrow><mfrac><mn>180</mn><pi/></mfrac><mo>*</mo><mi>angle</mi><mo>(</mo><mi>T</mi><mo>,</mo><mn>3</mn><mo>)</mo></mrow></mfenced></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mfenced><mrow><mi>A1</mi><mo>+</mo><mi>B1</mi><mo>+</mo><mi>C1</mi><mo>=</mo><mn>180</mn></mrow></mfenced></apply></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>A1</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>B1</mi><csymbol definitionURL="http://.../units/degree/angular">°</csymbol></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mi>C1</mi><csymbol definitionURL="http://.../units/degree/angular">°</csymbol></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mi>A1</mi><mo>,</mo><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>4</mn><mo>,</mo><mn>4</mn></mrow></mfenced><mo>,</mo><mn>81</mn><mo>,</mo><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>,</mo><mi>B1</mi><mo>,</mo><mi>d2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mn>4</mn></mrow></mfenced><mo>,</mo><mn>45</mn><mo>,</mo><msqrt><mn>37</mn></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>,</mo><mi>C1</mi><mo>,</mo><mi>d3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mo>-</mo><mn>2</mn></mrow></mfenced><mo>,</mo><mn>54</mn><mo>,</mo><mn>6</mn><mo>*</mo><msqrt><mn>2</mn></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>,</mo><mi>sol2</mi><mo>,</mo><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>81</mn><mo> </mo><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>,</mo><mn>45</mn><mo> </mo><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>,</mo><mn>54</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>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"><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><mspace linebreak="newline"/><mi>C</mi><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> N'hi ha prou amb calcular l'angle entre els 3 vectors, agafats de 2 en 2:
A és el vèrtex que correspon als vectors «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 mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mfenced mathcolor=¨#000066¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»AB«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»AB«/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¨»AC«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mfenced mathcolor=¨#000066¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»AC«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»AC«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
B és el vèrtex que correspon als vectors «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mover mathcolor=¨#000066¨»«mi mathvariant=¨bold¨»BA«/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»«mfenced mathcolor=¨#000066¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»B«/mi»«mi mathvariant=¨bold-italic¨»A«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»BA«/mi»«mn mathvariant=¨bold¨»12«/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»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mfenced mathcolor=¨#000066¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»B«/mi»«mi mathvariant=¨bold-italic¨»C«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»B«/mi»«mi mathvariant=¨bold-italic¨»C«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
C és el vèrtex que correspon als vectors «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mover mathcolor=¨#000066¨»«mi mathvariant=¨bold¨»CA«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mfenced mathcolor=¨#000066¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»C«/mi»«mi mathvariant=¨bold-italic¨»A«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»C«/mi»«mi mathvariant=¨bold-italic¨»A«/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¨»CB«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mfenced mathcolor=¨#000066¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»C«/mi»«mi mathvariant=¨bold-italic¨»B«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»C«/mi»«mi mathvariant=¨bold-italic¨»B«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math».

L'angle entre els vectors es calcula amb:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfenced mathcolor=¨#00007F¨»«mover»«mrow»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨»,«/mo»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mrow»«mo mathvariant=¨bold¨»^«/mo»«/mover»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»acos«/mi»«mfrac mathcolor=¨#00007F¨»«mrow»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨»§#183;«/mo»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mrow»«mrow»«mfenced open=¨||¨ close=¨||¨»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mfenced»«mo mathvariant=¨bold¨»§#183;«/mo»«mfenced open=¨||¨ close=¨||¨»«mover»«msub»«mi mathvariant=¨bold¨»v«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«/mfenced»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»acos«/mi»«mfrac mathcolor=¨#00007F¨»«mrow»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»y«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«msub»«mi mathvariant=¨bold¨»y«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mrow»«mrow»«msqrt»«msup»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«msub»«mi mathvariant=¨bold¨»y«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«mo mathvariant=¨bold¨»§#183;«/mo»«msqrt»«msup»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«msub»«mi mathvariant=¨bold¨»y«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»

Cal comprovar que sumin 180º
La situació és #G1
Els angles d'un triangle són tots positius!!

]]> Per exemple, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»acos«/mi»«mfrac mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»AB«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»AC«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»A«/mi»«mi mathvariant=¨bold-italic¨»B«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»A«/mi»«mi mathvariant=¨bold-italic¨»C«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«msqrt»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»A«/mi»«mi mathvariant=¨bold-italic¨»B«/mi»«msup»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»AB«/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¨»AC«/mi»«msup»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»AC«/mi»«msup»«mn mathvariant=¨bold¨»2«/mn»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mrow»«/mfrac»«/mstyle»«/math»

]]> 1MA.05.4.50DT Distància entre dos punts

Distància entre dos punts

És el mòdul del vector que els uneix:

]]> 0.0000000 0.0000000 0 1MA.05.4.51Q Distància entre dos punts Quina és la distància entre els punts «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»p«/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¨»p«/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¨»Q«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»q«/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¨»q«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math»?

Format de la resposta: radical simplificat

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]]> 1MA.05.4.60DT Distància d'un punt a una recta

Distància d'un punt a una recta
És la longitud de la perpendicular traçada des del punt fins a la recta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»d«/mi»«mo mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»P«/mi»«mo mathcolor=¨#003300¨»,«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»r«/mi»«mo mathcolor=¨#003300¨»)«/mo»«mo mathcolor=¨#003300¨»§#160;«/mo»«mo mathcolor=¨#003300¨»=«/mo»«mo mathcolor=¨#003300¨»§#160;«/mo»«mfrac mathcolor=¨#003300¨»«mfenced open=¨\|¨ close=¨\|¨»«mrow»«msub»«mi mathvariant=¨bold¨»ax«/mi»«mi mathvariant=¨bold¨»P«/mi»«/msub»«mo»+«/mo»«msub»«mi mathvariant=¨bold¨»by«/mi»«mi mathvariant=¨bold¨»P«/mi»«/msub»«mo»+«/mo»«mi mathvariant=¨bold¨»c«/mi»«/mrow»«/mfenced»«msqrt»«msup»«mi mathvariant=¨bold¨»a«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi mathvariant=¨bold¨»b«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«/math»
Exemple, amb P(1,2) i r una recta d'equació és 3x-5y+6=0: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold-italic¨»d«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold-italic¨»P«/mi»«mo mathvariant=¨bold¨»,«/mo»«mi mathvariant=¨bold-italic¨»r«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»=«/mo»«mfrac»«mfenced open=¨\|¨ close=¨\|¨»«mrow»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»6«/mn»«/mrow»«/mfenced»«msqrt»«msup»«mn mathvariant=¨bold¨»3«/mn»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«mfenced»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»5«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mfrac»«mo mathvariant=¨bold¨»=«/mo»«mfrac»«mn mathvariant=¨bold¨»4«/mn»«msqrt»«mn mathvariant=¨bold¨»34«/mn»«/msqrt»«/mfrac»«mo mathvariant=¨bold¨»=«/mo»«mfrac»«mrow»«mn mathvariant=¨bold¨»2«/mn»«msqrt»«mn mathvariant=¨bold¨»34«/mn»«/msqrt»«/mrow»«mn mathvariant=¨bold¨»17«/mn»«/mfrac»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold-italic¨»u«/mi»«mo mathvariant=¨bold¨».«/mo»«mi mathvariant=¨bold-italic¨»d«/mi»«mo mathvariant=¨bold¨».«/mo»«/mrow»«/mstyle»«/math»
Si no recordeu l'expressió, la podeu retrobar cercant el punt d'intersecció Q entre la recta r i la recta perpendicular a r que passa per P. Trobat Q, es pot calcular el mòdul del vector PQ

]]> 0.0000000 0.0000000 0 1MA.05.4.61Q Distància d'un punt a una recta Quina és la distància entre el punt «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»P«/mi»«mfenced mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p_«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math» i la recta r d'equació «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»e_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mstyle»«/math»?

Format de la resposta: radical simplificat

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«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»d«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»P«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»,«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»r«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mfrac mathcolor=¨#000066¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q_«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p_«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q_«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p_«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q_«/mi»«mn mathvariant=¨bold¨»3«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfenced»«msqrt»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q_«/mi»«mn mathvariant=¨bold¨»1«/mn»«msup»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mfrac»«/mrow»«/mstyle»«/math»

]]> 1MA.05.4.70DT Distància entre rectes paral·leles

Distància entre rectes paral·leles

S'agafa un punt de l'una i es calcula la seva distància a l'altra:

Si r≡ax+by+c=0 i S és un punt de s:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»d«/mi»«mo mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»s«/mi»«mo mathcolor=¨#003300¨»,«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»r«/mi»«mo mathcolor=¨#003300¨»)«/mo»«mo mathcolor=¨#003300¨»§#160;«/mo»«mo mathcolor=¨#003300¨»=«/mo»«mo mathcolor=¨#003300¨»§#160;«/mo»«mfrac mathcolor=¨#003300¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«msub»«mi mathvariant=¨bold¨»ax«/mi»«mi mathvariant=¨bold¨»S«/mi»«/msub»«mo»+«/mo»«msub»«mi mathvariant=¨bold¨»by«/mi»«mi mathvariant=¨bold¨»S«/mi»«/msub»«mo»+«/mo»«mi mathvariant=¨bold¨»c«/mi»«/mrow»«/mfenced»«msqrt»«msup»«mi mathvariant=¨bold¨»a«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi mathvariant=¨bold¨»b«/mi»«mn»2«/mn»«/msup»«/msqrt»«/mfrac»«/math»

]]> 0.0000000 0.0000000 0 1MA.05.4.71Q Distància entre dues rectes paral·leles Quina és la distància entre les dues rectes d'equacions
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»r«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8801;«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u_«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»n«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/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»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»s«/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¨»2«/mn»«/mrow»«/mstyle»«/math»?

Format de la resposta: radical simplificat

]]>

]]> 1.0000000 0.5000000 0 0 #r_55]]> <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 definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>p_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>p_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>u_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>u_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>e_1</mi><mo>=</mo><mfrac><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>-</mo><mi>p_1</mi></mtd></mtr></mtable></mfenced><mi>u_1</mi></mfrac><mo>=</mo><mfrac><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>y</mi><mo>-</mo><mi>p_2</mi></mtd></mtr></mtable></mfenced><mi>u_2</mi></mfrac></mtd></mtr><mtr><mtd><mi>n1</mi><mo>=</mo><mi>x</mi><mo>-</mo><mi>p_1</mi></mtd></mtr><mtr><mtd><mi>n2</mi><mo>=</mo><mi>y</mi><mo>-</mo><mi>p_2</mi></mtd></mtr><mtr><mtd><mi>q_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>q_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>q_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>e_2</mi><mo>=</mo><mi>q_1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>q_2</mi><mo>*</mo><mi>y</mi><mo>+</mo><mi>q_3</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>rang</mi><mfenced><mtable><mtr><mtd><mi>u_1</mi></mtd><mtd><mo>-</mo><mi>q_2</mi></mtd></mtr><mtr><mtd><mi>u_2</mi></mtd><mtd><mi>q_1</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>1</mn></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>q21</mi><mo>=</mo><mo>-</mo><mi>q_2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_55</mi><mo>=</mo><mi>distància</mi><mo>(</mo><mi>punt</mi><mo>(</mo><mi>p_1</mi><mo>,</mo><mi>p_2</mi><mo>)</mo><mo>,</mo><mi>q_1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>q_2</mi><mo>*</mo><mi>y</mi><mo>+</mo><mi>q_3</mi><mo>=</mo><mn>0</mn><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>p_1</mi><mo>,</mo><mi>p_2</mi><mo>,</mo><mi>u_1</mi><mo>,</mo><mi>u_2</mi><mo>,</mo><mi>q_1</mi><mo>,</mo><mi>q_2</mi><mo>,</mo><mi>q_3</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>,</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>,</mo><mo>-</mo><mn>6</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>9</mn></math></output></command><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"><mfrac><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mn>1</mn></mfrac><mo>=</mo><mfrac><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>y</mi><mo>+</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mn>6</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_55</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>26</mn><mo>*</mo><msqrt><mn>37</mn></msqrt></mrow><mn>37</mn></mfrac></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>r</mi><mo>_</mo><mn>55</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">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Primer cal esbrinar si són paral·leles, calculant el determinant amb els vectors directors: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced mathcolor=¨#000066¨ open=¨|¨ close=¨|¨»«mtable»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u_«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mtd»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q«/mi»«mn mathvariant=¨bold¨»21«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»u_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mtd»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q_«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«/mstyle»«/math»

Després s'aplica l'expressió de la distància d'un punt de l'una R(#p_1,#p_2) a l'altra, s:#e_2:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»d«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»R«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»,«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»s«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mfrac mathcolor=¨#000066¨»«mfenced open=¨|¨ close=¨|¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q_«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»§#183;«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p_«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»§#183;«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»+«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q_«/mi»«mn mathvariant=¨bold¨»3«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«msqrt»«msup»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q_«/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¨»q_«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mfrac»«/mrow»«/mstyle»«/math»

]]> 1MA.05.4.80DT Com es treu un valor absolut

Treure valor absolut

Com que: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfenced mathcolor=¨#003300¨ open=¨|¨ close=¨|¨»«mi mathvariant=¨bold¨»a«/mi»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfenced mathcolor=¨#003300¨ open=¨{¨ close=¨¨»«mtable»«mtr»«mtd»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»si«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»§#62;«/mo»«mn mathvariant=¨bold¨»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»si«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»§lt;«/mo»«mn mathvariant=¨bold¨»0«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«/mrow»«/mstyle»«/math»

quan traiem el valor absolut de l'equació |ax+by+c|=d, apareixen dues equacions (i 2 solucions possibles):

ax+by+c = d
ax+by+c = -d

]]> 0.0000000 0.0000000 0 1MA.05.4.81Q Recta // a distància fixa(rac) Troba l'equació cartesiana d'una recta paral·lela a la recta r 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» i que en sigui distant de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»v«/mi»«/mrow»«/mstyle»«/math» unitats.

Atenció: pot ser que hi hagi més d'una solució

Format de la resposta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«mo»-«/mo»«mn»5«/mn»«mo»=«/mo»«mn»0«/mn»«mo»,«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«mo»+«/mo»«mn»7«/mn»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math»

]]> Solucions en verd: #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 definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>d</mi><mo>=</mo><mo>-</mo><mi>c</mi></mtd></mtr><mtr><mtd><mi>P</mi><mo>=</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>-</mo><mi>c</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>e_1</mi><mo>=</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>b</mi><mo>*</mo><mi>y</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>e_2</mi><mo>=</mo><mi>a</mi><mo>*</mo><mn>0</mn><mo>+</mo><mi>b</mi><mo>*</mo><mi>y</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd/></mtr><mtr><mtd><mi>r_50</mi><mo>=</mo><mfrac><mfenced close="|" open="|"><mrow><mo>-</mo><mi>c</mi><mo>+</mo><mi>m</mi></mrow></mfenced><msqrt><mrow><msup><mfenced><mi>a</mi></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mfrac></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><msqrt><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr><mtr><mtd><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mfrac><mfenced close="|" open="|"><mrow><mo>-</mo><mi>c</mi><mo>+</mo><mi>m</mi></mrow></mfenced><msqrt><mrow><msup><mfenced><mi>a</mi></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mfrac><mo>=</mo><msqrt><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></mfenced></mtd></mtr><mtr><mtd><mi>r_1</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>r_2</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>r_1</mi><mo>∈</mo><rationals/></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>a</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>r_1</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>r_2</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></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>recta</mi><mo>(</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>b</mi><mo>*</mo><mi>y</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></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><mo>(</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>b</mi><mo>*</mo><mi>y</mi><mo>+</mo><mi>r_1</mi><mo>=</mo><mn>0</mn></mrow></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</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><mo>(</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>b</mi><mo>*</mo><mi>y</mi><mo>+</mo><mi>r_2</mi><mo>=</mo><mn>0</mn></mrow></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>q_1</mi><mo>,</mo><mi>q_2</mi><mo>,</mo><mi>q_3</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q_1</mi><mo>,</mo><mi>q_2</mi><mo>,</mo><mi>q_3</mi></math></output></command><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"><mn>8</mn><mo>*</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_50</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msqrt><mn>65</mn></msqrt><mo>*</mo><mfenced close="|" open="|"><mrow><mi>m</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mrow><mn>65</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>65</mn></msqrt></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"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>m</mi><mo>=</mo><mo>-</mo><mn>62</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>m</mi><mo>=</mo><mn>68</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>62</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>68</mn></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 close="}" open="{"><mtable align="center"><mtr><mtd><mn>8</mn><mo>*</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>62</mn><mo>=</mo><mn>0</mn><mo>,</mo><mn>8</mn><mo>*</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>68</mn><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced></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">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> La recta paral·lela té el mateix vector director i,
per tant, la seva equació és de la forma #a x + y + c = 0.
Un punt de la recta que ens donen és (0,#d) que es troba substituint x per 0: #e_2 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8658;«/mo»«mi»y«/mi»«mo»=«/mo»«/math»d
Es calcula la distància d'aquest punt (0, #d) a #a x + y + c = 0 i s'iguala a #v.
Al treure els valors absoluts, surten dues possibilitats pels signes, per tant dues equacions.

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