1MA 06. LLOCS GEOMÈTRICS/1MA.06.2 Paral·lelogSimetries 1MA.06.2.11Q EqCostatParal·lelogram Considera els punts «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»C«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math». A partir d'aquests punts, es pot construir el paral·lelogram ABCD.

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

Format:

AB= paramètriques {x=k+1,y=k-1}

CD=explícita

]]> #G2

]]> 1.0000000 0.3333333 0 0 AB=#sol1CD=#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mi style="color:#ffc800">variables</mi><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>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>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>c1</mi><mo>,</mo><mi>c2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>v1</mi><mo>=</mo><mi>b1</mi><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>v2</mi><mo>=</mo><mi>b2</mi><mo>-</mo><mi>a2</mi></mtd></mtr><mtr><mtd><mi>u1</mi><mo>=</mo><mi>c1</mi><mo>-</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>u2</mi><mo>=</mo><mi>c2</mi><mo>-</mo><mi>a2</mi></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>a1</mi><mo>≠</mo><mi>b1</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>a2</mi><mo>≠</mo><mi>b2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mfrac><mi>v2</mi><mi>v1</mi></mfrac><mo>∈</mo><integers/></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>distància</mi><mfenced><mrow><mi>B</mi><mo>,</mo><mi>C</mi></mrow></mfenced><mo>></mo><mn>3</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>distància</mi><mfenced><mrow><mi>A</mi><mo>,</mo><mi>C</mi></mrow></mfenced><mo>></mo><mn>3</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>u1</mi><mo>*</mo><mi>v2</mi><mo>≠</mo><mi>u2</mi><mo>*</mo><mi>v1</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math 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align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>4</mn><mo>*</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>4</mn><mo>*</mo><mi>k</mi><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler2</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>C</mi><mi>D</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="inputCompound">true</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> a) L'equació de la recta AB és l'equació d'una recta que passa pel punt A «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math» i que té per vector director el vector «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mover mathcolor=¨#00007F¨»«mi mathvariant=¨bold¨»AB«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«/mover»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»vr«/mi»«/mstyle»«/math» .

La recta és la recta vermella: #G1

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

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

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