Question

Dados os pontos A(-1, -1), B(5, -7) e C(x, 2), determine x sabendo que C é equidistante dos pontos A e B.

Answer 1

A distancia entre dois pontos é dada por:

$Distancia_{a,b}~=~\sqrt{\left(x_a-x_b\right)^2~+~\left(y_a-y_b\right)^2}$

Se C é equidistante de A e B, então podemos igualar a distancia AC à distancia BC:

    $\begin{array}{rcl}Distancia_{A,C}&=&Distancia_{B,C}\\\\\\\sqrt{\left(x_A-x_C\right)^2~+~\left(y_A-y_C\right)^2}&=&\sqrt{\left(x_B-x_C\right)^2~+~\left(y_B-y_C\right)^2}\\\\\\\left(x_A-x_C\right)^2~+~\left(y_A-y_C\right)^2&=&\left(x_B-x_C\right)^2~+~\left(y_B-y_C\right)^2\end{array}$

Substituindo os dados:

      $\begin{array}{rcl}\left(-1-x\right)^2~+~\left(-1-2\right)^2&=&\left(5-x\right)^2~+~\left(-7-2\right)^2\\\\\\\left(x^2+2x+1\right)~+~\left(-3\right)^2&=&\left(x^2-10x+25\right)~+~\left(-9\right)^2\\\\\\x^2+2x+1+9&=&x^2-10x+25~+~81\\\\\\x^2+2x-x^2+10x&=&106-10\\\\\\12x&=&96\\\\\\x&=&\dfrac{96}{12}\\\\\\&\boxed{x~=~8}&\end{array}$