Question

14) Determine o valor de y para que o triângulo de vértices A( 5, y), B( -4, 3) e C( -1, -2) tenha área igual a 25.

Answer 1

$\boxed{\begin{array}{c}\tt{\underline{\acute{A}rea~do~tri\hat{a}ngulo~em~func_{\!\!,}\tilde{a}o}}\\\sf{de~suas~coordenadas}\end{array}}\\\sf{A(x_A, y_A)~B(x_B,y_B)~C(x_C,y_C)}\\\sf{M}=\begin{bmatrix}x_A&y_A&1\\x_B&y_B&1\\x_C&y_C&1\end{bmatrix}\\\sf{A=\dfrac{|det~M|}{2}}$

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$\sf{M}=\begin{bmatrix}5&y&1\\-4&3&1\\-1&-2&1\end{bmatrix}$

$\sf{det~M=5(3+2)-y(-4+1)+1(8+3)}\\\sf{det~M=25+3y+11}\\\sf{det~M=36+3y}$

$\sf{A=\dfrac{|36+3y|}{2}}\\\sf{25=\dfrac{|36+3y|}{2}}\\\sf{|36+3y|=50}\\\sf{36+3y=50}\\\sf{3y=50-36}\\\sf{3y=14}\\\sf{y=\dfrac{14}{3}}\\\sf{36+3y=-50}\\\sf{3y=-36-50}\\\sf{3y=-86}\\\sf{y=-\dfrac{86}{3}}$