We have to find the equation of the perpendicular bisector of the segment whose endpoints are $P (5, 2)$ and $Q (7, 4) .$ We will do this in three steps.

Find the midpoint of $\overline{P Q} .$
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 $\overline{P Q}$

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

M

of

\overline{P Q} .

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

To find the coordinates of the midpoint

M,

we will substitute

(5, 2)

and

(7, 4)

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 $(5, 2)$ & $(7, 4)$

M (\frac{5 + 7}{2}, \frac{2 + 4}{2})

AddTerms

Add terms

M (\frac{1 2}{2}, \frac{6}{2})

CalcQuot

Calculate quotient

M (6, 3)

Therefore,

M (6, 3)

is the midpoint of

\overline{P Q} .

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{P Q} .

To do so, we will use the Slope Formula.

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

Let's substitute

(5, 2)

and

(7, 4)

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 $(5, 2)$ & $(7, 4)$

m = \frac{4 - 2}{7 - 5}

SubTerms

Subtract terms

m = \frac{2}{2}

CalcQuot

Calculate quotient

m = 1

The slope of

\overline{P Q}

is

1 .

Let

m_{p}

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

- 1 .

1 \cdot m_{p} = - 1 \Leftrightarrow m_{p} = - 1

The slope of the perpendicular bisector of

\overline{P Q}

is

- 1 .

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

- 1

for

m

and

(6, 3)

for

(x_{1}, y_{1})

in the formula.

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

SubstituteValues

Substitute values

y - 3 = - 1 (x - 6)

Distr

Distribute $- 1$

y - 3 = - x + 6

AddEqn

$LHS + 3 = RHS + 3$

y = - x + 9

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

P (5, 2)

and

Q (7, 4)

is

y = - x + 9 .

Midpoint of PQ​

Slope of the Perpendicular Bisector

Equation of the Perpendicular Bisector

Midpoint of $\overline{P Q}$

Exercise