Question

Find an

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

Answer 1

`y=(2)/(3)x+3`

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=(y_2-y_1)/(x_2-x_1)`

`\displaystyle m=(4-(-2))/(-5-(-1))=(6)/(-4)=-(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=-(1)/(m_1)`

`\displaystyle m_2=-(1)/(-(3)/(2))=(2)/(3)`

The equation of the perpendicular bisector has the form:

`y=(2)/(3)x+b`

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

`\displaystyle \bar x=(-1-5)/(2)=-3`

`\displaystyle \bar y=(4-2)/(2)=1`

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

`1=(2)/(3)(-3)+b`

`b=1+2=3`

Thus, the equation for the perpendicular bisector is

`\boxed{y=(2)/(3)x+3}`