Question

Line through the points $(–2,6)\text{and}(4,8)$ is perpendicular to the line through the points $(8,12)\text{and}(x,24)$. Find the value of $x$.

Answer 1

Step $1:$ Finding slopes of given lines
Let the given points be
$A(-2,6),B(4,8),C(8,12)\text{and}D(x,24)$
We know that slope of a line through the points $(x_1,y_1),(x_2,y_2)$ is $m=\frac{y_2-y_1}{x_2-x_1}$
Here, $x_1=-2,y_1=6,x_2=4,y_2=8$
Slope of $AB=\frac{8-6}{4-(-2)}=\frac{2}{4+2}=\frac{2}{6}=\frac{1}{3}$

Slope of line $CD$ passing through
$C(8,12)\text{and}D(x,24)$
Here $x_1=8,y_1=12,x_2=x,y_2=24$
Slope of $CD=\frac{24-12}{x-8}=\frac{12}{x-8}$

Step $2:$ Solve for value of $x$
We know that if two lines are perpendicular, then the product of their slopes is $-1$
So, Slope of $AB\times$ Slope of $CD=-1$
$\Rightarrow \frac{1}{3}\times \left(\frac{12}{x-8}\right)=-1$
$\Rightarrow \frac{4}{(x-8)}=-1$
$\Rightarrow 4=-(x-8)$
$\Rightarrow x=4$

Final Answer :
Therefore, the value of $x\text{is}4$