Question

calcule o valor de x e determine a medida das diagonais de cada um dos losangos​

Answer 1

A área de um losango é dada pela metade do produto das suas diagonais:

$\boxed{Area_{\diamondsuit}~=~\dfrac{d\cdot D}{2}}$

a)

São dados:

$\left\{\begin{array}{ccl}Area_{\diamondsuit}&=&9~cm^2\\d&=&(3x-6)~cm\\D&=&6~cm\end{array}\right$

Substituindo, temos

$9~=~\dfrac{(3x-6)\cdot6}{2}\\\\\\9~=~\dfrac{3x\cdot6-6\cdot6}{2}\\\\\\2\cdot9~=~18x-36\\\\\\-18x~=~-36-18\\\\\\-18x~=~-54\\\\\\x~=~\dfrac{-54}{-18}\\\\\\\boxed{x~=~3~cm}$

Assim, a digonal restante (d) vale:

$d~=~3x-6\\\\d~=~3\cdot 3-6\\\\d~=~9-6\\\\\boxed{d~=~3~cm}$

b)

São dados:

$\left\{\begin{array}{ccl}Area_{\diamondsuit}&=&10~cm^2\\d&=&4~cm\\D&=&\left(\dfrac{x}{3}+1\right)~cm\end{array}\right$

Substituindo, temos

$10~=~\dfrac{4\cdot\left(\dfrac{x}{3}+1\right)}{2}\\\\\\2\cdot10~=~4\cdot\left(\dfrac{x}{3}+1\right)\\\\\\20~=~4\cdot\dfrac{x}{3}+4\cdot 1\\\\\\-\dfrac{4x}{3}~=~4~-~20\\\\\\-\dfrac{4x}{3}~=~-16\\\\\\x~=~-16\cdot\left(-\dfrac{3}{4}\right)\\\\\\\boxed{x~=~12~cm}$

Assim, a digonal restante (D) vale:

$D~=~\dfrac{x}{3}+1\\\\\\D~=~\dfrac{12}{3}+1\\\\\\D~=~4+1\\\\\\\boxed{D~=~5~cm}$

c)

São dados:

$\left\{\begin{array}{ccl}Area_{\diamondsuit}&=&15~cm^2\\d&=&(3x-9)~cm\\D&=&5~cm\end{array}\right$

Substituindo, temos

$15~=~\dfrac{(3x-9)\cdot 5}{2}\\\\\\15~=~\dfrac{3x\cdot5-9\cdot5}{2}\\\\\\2\cdot15~=~15x-45\\\\\\-15x~=~-45-30\\\\\\-15x~=~-75\\\\\\x~=~\dfrac{-75}{-15}\\\\\\\boxed{x~=~5~cm}$

Assim, a digonal restante (d) vale:

$d~=~3x-9\\\\d~=~3\cdot 5-9\\\\d~=~15-9\\\\\boxed{d~=~6~cm}\\\\\\\Huge{\begin{array}{c}\Delta \tt{\!\!\!\!\!\!\,\,o}\!\!\!\!\!\!\!\!\:\,\perp\end{array}}Qualquer~d\acute{u}vida,~deixe~ um~coment\acute{a}rio$