1MA 02. RESOLUCIÓ DE TRIANGLES/1MA.02.3 Resolució de problemes 1MA.02.3.44Q 2003J TCosinus Triangle, àrea, suma costats D'un triangle sabem que la suma de dos dels costats és de #s cm, que l'angle C oposat al tercer costat val 30º  i que l'àrea és de #p cm2. 

Calculeu la longitud dels 3 costats i els 3 angles.

Format de les respostes: costats {2,3,4} angles {30,80,70}. Arrodonits a la unitat

]]> El triangle proposat és #G1

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Sabem que a+b = #s, per tant, b = #s - a.

L'altura sobre el costat a és h =  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»b«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#000066¨»sin«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#000066¨»30«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#186;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»§#160;«/mo»«mfrac mathcolor=¨#000066¨»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mfrac mathcolor=¨#000066¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«/mstyle»«/math».

L'àrea del triangle és doncs «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfrac mathcolor=¨#000066¨»«mstyle displaystyle=¨true¨»«mfrac»«mrow»«mi mathvariant=¨bold¨»a«/mi»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/mfenced»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«/mstyle»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#000066¨»p«/mi»«/mstyle»«/math» 

a · (#s - a) = 4 · #p té dues solucions intercanviables que són a i b.

 

També es pot raonar amb les 3 equacions i substituir a i h a la 3a per trobar b

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced mathcolor=¨#00007F¨ open=¨{¨ close=¨}¨»«mtable»«mtr»«mtd»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»b«/mi»«mo mathvariant=¨bold¨»=«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#8658;«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»=«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»b«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»b«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mn mathvariant=¨bold¨»30«/mn»«mo mathvariant=¨bold¨»§#186;«/mo»«mo mathvariant=¨bold¨»=«/mo»«mfrac»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»h«/mi»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨»=«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#8658;«/mo»«mfrac mathcolor=¨#00007F¨»«mrow»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»b«/mi»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»§#183;«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«/mstyle»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»p«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#8658;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»-«/mo»«msup mathcolor=¨#00007F¨»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»s«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»b«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»4«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»p«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»0«/mn»«/mstyle»«/math»

el costat c es pot calcular aplicant el teorema del cosinus

 

 Els 3 costats són #sol1

 

]]> Per a determinar l'angle A, com que sabem els 3 costats, apliquem el teorema del cosinus, aïllant l'angle A:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold-italic¨ mathcolor=¨#000066¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#000066¨»=«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#000066¨»a«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#000066¨»c«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#000066¨»o«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#000066¨»s«/mi»«mfenced mathcolor=¨#000066¨»«mfrac»«mrow»«msup»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»+«/mo»«msup»«mi mathvariant=¨bold¨»c«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»b«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»c«/mi»«/mrow»«/mfrac»«/mfenced»«/mrow»«/mstyle»«/math»

L'angle B és 180º - 30º - A.

I els angles són #sol2

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