Question

Find an
equation for the perpendicular bisector of the line segment whose endpoints
are (-1, -2) and (-5,4).

Answer 1

Answer:

$y=\frac{2}{3}x+3$

Step-by-step explanation:

Perpendicular Bisector

It's defined as a line that divides another line into two equal parts. The bisector passes through the midpoint of the line forming any angle, but if that angle is exactly 90°, then the bisector is also perpendicular.

We need to find the equation of the line that divides into equal parts the line with endpoints (-1, -2) and (-5,4) and is perpendicular to it.

First, let's find the slope of the line segment. The slope can be calculated with the formula:

$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$

$\displaystyle m=\frac{4-(-2)}{-5-(-1)}=\frac{6}{-4}=-\frac{3}{2}$

The line for the perpendicular bisector has a slope m2. Two perpendicular lines with slopes m1 and m2 must comply:

$m_1.m_2=-1$

Solving for m2:

$\displaystyle m_2=-\frac{1}{m_1}$

$\displaystyle m_2=-\frac{1}{-\frac{3}{2}}=\frac{2}{3}$

The equation of the perpendicular bisector has the form:

$y=\frac{2}{3}x+b$

Now we find the coordinates of the midpoint of the segment:

$\displaystyle \bar x=\frac{-1-5}{2}=-3$

$\displaystyle \bar y=\frac{4-2}{2}=1$

The midpoint is (-3,1). Using this point will allow us to find the value of b:

$1=\frac{2}{3}(-3)+b$

$b=1+2=3$

Thus, the equation for the perpendicular bisector is

$\boxed{y=\frac{2}{3}x+3}$