Exercise 56 Page 351

We have to find the equation of the perpendicular bisector of the segment whose endpoints are $D (- 2, - 4)$ and $E (2, 1) .$ We will do this in three steps.

Find the midpoint of $\overline{D E} .$
Find the slope of the perpendicular bisector.
Use the point-slope form to write the equation of the line.

Let's go for it!

Midpoint of $\overline{D E}$

A segment bisector contains the midpoint of the segment. We will use the Midpoint Formula to find the midpoint

M

of

\overline{D E} .

M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

To find the coordinates of the midpoint

M,

we will substitute

(- 2, - 4)

and

(2, 1)

for

(x_{1}, y_{1})

and

(x_{2}, y_{2})

in the formula.

M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

SubstitutePoints

Substitute $(- 2, - 4)$ & $(2, 1)$

M (\frac{- 2 + 2}{2}, \frac{- 4 + 1}{2})

AddTerms

Add terms

M (\frac{0}{2}, \frac{- 3}{2})

CalcQuot

Calculate quotient

M (0, \frac{- 3}{2})

MoveNegNumToFrac

Put minus sign in front of fraction

M (0, - \frac{3}{2})

Therefore,

M (0, - \frac{3}{2})

is the midpoint of

\overline{D E} .

Slope of the Perpendicular Bisector

A perpendicular bisector is perpendicular to the segment through the midpoint. In order to find the slope of the bisector, we will first find the slope of

\overline{D E} .

To do so, we will use the Slope Formula.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Let's substitute

(- 2, - 4)

and

(2, 1)

for

(x_{1}, y_{1})

and

(x_{2}, y_{2})

in this formula.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(- 2, - 4)$ & $(2, 1)$

m = \frac{1 - ( - 4 )}{2 - ( - 2 )}

SubNeg

$a - (- b) = a + b$

m = \frac{5}{4}

The slope of

\overline{D E}

is

\frac{5}{4} .

Let

m_{p}

be the slope of the perpendicular bisector. The product of the slopes of two perpendicular lines is

- 1 .

\frac{5}{4} \cdot m_{p} = - 1 \Leftrightarrow m_{p} = - \frac{4}{5}

The slope of the perpendicular bisector of

\overline{D E}

is

- \frac{4}{5} .

Equation of the Perpendicular Bisector

Since we know a point and the slope of the bisector, we will use the point-slope form of a line to write its equation.

y - y_{1} = m (x - x_{1})

Let's substitute

- \frac{4}{5}

for

m

and

(0, - \frac{3}{2})

for

(x_{1}, y_{1})

in the formula.

y - y_{1} = m (x - x_{1})

SubstituteValues

Substitute values

y - (- \frac{3}{2}) = - \frac{4}{5} (x - 0)

▼

Solve for

y

SubNeg

$a - (- b) = a + b$

y + \frac{3}{2} = - \frac{4}{5} (x - 0)

Distr

Distribute $- \frac{4}{5}$

y + \frac{3}{2} = - \frac{4}{5} x + 0

IdPropAdd

Identity Property of Addition

y + \frac{3}{2} = - \frac{4}{5} x

SubEqn

$LHS - \frac{3}{2} = RHS - \frac{3}{2}$

y = - \frac{4}{5} x - \frac{3}{2}

The equation of the perpendicular bisector of the segment whose endpoints are

D (- 2, - 4)

and

E (2, 1)

is

y = - \frac{4}{5} x - \frac{3}{2} .