Question:
If A(2, -5), B(-2, 5), C(x, 3) and D(1, 1) be four points such that AB and CD are perpendicular to each other, find the value of x.
Solution:
For two lines to be perpendicular, their product of slope must be equal to -1.
Given points are A(2, -5),B(-2, 5) and C(x, 3),D(1, 1)
slope $=\left(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\right)$
$\Rightarrow$ Slope of line $A B$ is equal to
$\left(\frac{5+5}{-2-2}\right)$
$=\left(\frac{10}{-4}\right)$
$=\left(\frac{-5}{2}\right)$
$=-2.5$
And the slope of line CD is equal to
$\left(\frac{1-3}{1-x}\right)$
$=\left(\frac{-2}{1-x}\right)$
Their product must be equal to -1
the slope of line AB×Slope of line CD = -1
$\Rightarrow-2.5 \times\left(\frac{-2}{1-x}\right)=-1 \Rightarrow 5=x-1$
⇒x = 6