Question

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

• A. $2$
• B. $4$
• C. $1$
• D. $8$

Answer 1

The correct option is B $4$

Given, $A(-2,6),B(4,8)$, $C(8,12)$ and $D(x,24),$
To Find: The value of $x$.

Formula: Slope of the line passing through the points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2-y_1}{x_2-x_1}.$

Let the slope of line $AB$ be $m_1$ and the slope of line $CD$ be $m_2$.

$m_1=\frac{8-6}{4-(-2)}=\frac{2}{4+2}=\frac{2}{6}=\frac{1}{3}$

$m_2=\frac{24-12}{x-8}=\frac{12}{x-8}$

As $AB$ is perpendicular to $CD,$ then
$m_1\times m_2=-1$
$\Rightarrow \frac{1}{3}\times \frac{12}{x-8}=-1$
$\Rightarrow \frac{4}{x-8}=-1$
Cross multiply
$\Rightarrow 4=-(x-8)$
$\Rightarrow 4=-x+8$
$\Rightarrow x=8-4$
$\Rightarrow x=4$

Option (b.) is correct.