Question

Geometry:( will give brainist to correct answer:)

Answer 1

Given:

End points a line segment are (9,-8) and (-1,-4).

To find:

The equation of perpendicular bisector of given line.

Solution:

Slope of given line is

$m=\dfrac{y_2-y_1}{x_2-x_1}$

$m_1=\dfrac{-4-(-8)}{-1-9}$

$m_1=\dfrac{-4+8}{-10}$

$m_1=\dfrac{4}{-10}$

$m_1=-\dfrac{2}{5}$

Product of slopes of two perpendicular line is -1.

$m_1\times m_2=-1$

$(-\dfrac{2}{5})\times m_2=-1$

$m_2=\dfrac{5}{2}$

So, slope of perpendicular bisector is $\dfrac{5}{2}$.

Perpendicular bisector passes through the midpoint of given endpoints.

$Midpoint=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$

$Midpoint=\left(\dfrac{9+(-1)}{2},\dfrac{(-8)+(-4)}{2}\right)$

$Midpoint=\left(\dfrac{8}{2},\dfrac{-12}{2}\right)$

$Midpoint=\left(4,-6\right)$

So, perpendicular bisector passes through (4,-6) and having slope $\dfrac{5}{2}$. The equation of perpendicular bisector is

$y-y_1=m(x-x_1)$

where, m is slope.

$y-(-6)=\dfrac{5}{2}(x-4)$

$y+6=\dfrac{5}{2}(x)+\dfrac{5}{2}(-4)$

$y+6=\dfrac{5}{2}x-10$

$y=\dfrac{5}{2}x-10-6$

$y=\dfrac{5}{2}x-16$

Therefore, the equation of perpendicular bisector is $y=\dfrac{5}{2}x-16$.