Question

If the line through the points $(-2,6)$ and $(4,8)$ is perpendicular to the line through the points $(8,12)$ and $(x,24)$, then the value of $x$ is

Answer 1

Let $A(-2,6),B(4,8),C(8,12)$ and $D(x,24)$ be the given points.
Let $m_1$ and $m_2$ be the slopes of $AB$ and $CD$ respectively. Then,
$m_1=\text{slope of}AB=\frac{(8-6)}{4-(-2)}=\frac{2}{6}=\frac{1}{3}$
$m_2=\text{slope of}CD=\frac{(24-12)}{(x-8)}=\frac{12}{(x-8)}$
Now, $AB\perp CD\Rightarrow m_1{m}_2=-1$
$\Rightarrow \frac{1}{3}\times \frac{12}{(x-8)}=-1$
$\Rightarrow -x+8=4\Rightarrow x=4$