Question

URGENTEE
Os pontos A(x, 4), B(2,1) e C(-1, -2), formam um triângulo ABC com área igual a
4. Calcule x.

Answer 1

Explicação passo-a-passo:

Quando temos as coordenadas dos três vértices de um triângulo, utilizamos a seguinte fórmula para calcular a área:

$S=\dfrac{|D|}{2}\ \text onde\ D=\left|\begin{array}{ccc}x_{a} &y_{a} &1\\x_{b} &y_{b} &1\\x_{c} &y_{c} &1\end{array}\right|$

Temos que S=4, A(x, 4), B(2, 1) e C(-1, -2).

$4=\dfrac{|D|}{2}\\\\I)\boxed{|D|=8}$

Calculando o determinante D:

$\ D=\left|\begin{array}{ccc}x &4 &1\\2 &1 &1\\-1&-2 &1\end{array}\right|\\\\\text Usando\ a\ Regra\ de\ Sarrus:\\\\D=\left|\begin{array}{ccc}x &4 &1\\2 &1 &1\\-1&-2 &1\end{array}\right\left|\begin{array}{ccc}x &4 \\2 &1 \\-1&-2 \end{array}\right|\\\\D=x.1.1+4.1.(-1)+1.2.(-2)-4.2.1-x.1.(-2)-1.1.(-1)\\\\D=x-4-4-8+2x+1\\\\II)\boxed{D=3x-15}$

Substituindo $II$ na $I$:

$|3x-15|=8\\\\3x-15=8\\\\3x=8+15\\\\3x=23\\\\\boxed{\boxed{x=\dfrac{23}{3} }}\\\\ou\\\\3x-15=-8\\\\3x=-8+15\\\\3x=7\\\\\boxed{\boxed{x=\dfrac{7}{3}}}$