1. ( ) ( ) , , EXAMPLE 3 Find the midpoint of a line segment
Find the midpoint of the line segment joining (–5, 1) and (–1, 6).
SOLUTION
Let ( x1, y1 ) = (–5, 1) and ( x2, y2 ) = (–1, 6).
( )
x1 + x2 y1 + y2 2 ,
– 5 + (– 1)
( ) = 2 ,
= (– 3, ) 72

2. ( ) ( ) , , EXAMPLE 4 Find a perpendicular bisector
Write an equation for the perpendicular bisector of the line segment joining A(–3, 4) and B(5, 6).
SOLUTION
STEP 1
Find the midpoint of the line segment.
( )
x1 + x2 y1 + y2 2 , =
( ) – 2 ,
= (1, 5)

3. EXAMPLE 4
Find a perpendicular bisector
STEP 2
Calculate the slope of AB
m =
y2 – y1
x2 – x1 =
6 – 4
5 – (– 3) = 28 = 14
STEP 3
Find the slope of the perpendicular bisector: – 1m – =
1 1/4
= – 4

4. EXAMPLE 4
Find a perpendicular bisector
STEP 4
Use point-slope form:
y – 5 = – 4(x – 1),
y = – 4x + 9. or
ANSWER
An equation for the perpendicular bisector of AB is y = – 4x + 9.

5. EXAMPLE 5
Solve a multi-step problem
Asteroid Crater
Many scientists believe that an asteroid slammed into Earth about 65 million years ago on what is now Mexico’s Yucatan peninsula, creating an enormous crater that is now deeply buried by sediment. Use the labeled points on the outline of the circular crater to estimate its diameter. (Each unit in the coordinate plane represents 1 mile.)

6. Solve a multi-step problem
EXAMPLE 5
Solve a multi-step problem
SOLUTION
STEP 1
Write equations for the perpendicular bisectors of AO and OB using the method of Example 4.
y = –x + 34
Perpendicular bisector of AO
y = 3x
Perpendicular bisector of OB

7. Solve a multi-step problem
EXAMPLE 5
Solve a multi-step problem
STEP 2
Find the coordinates of the center of the circle, where AO and OB intersect, by solving the system formed by the two equations in Step 1.
y = –x + 34
Write first equation.
3x = –x + 34
Substitute for y.
4x = –76
Simplify.
x = –19
Solve for x.
y = –(–19) + 34
Substitute the x-value into the first equation.
y = 53
Solve for y.
The center of the circle is C (–19, 53).

8. Solve a multi-step problem
EXAMPLE 5
Solve a multi-step problem
STEP 3
Calculate the radius of the circle using the distance formula. The radius is the distance between C and any of the three given points.
Use (x1, y1) = (0, 0) and (x2, y2) = (–19, 53).
OC = (–19 – 0)2 + (53 – 0)2 =
ANSWER
The crater has a diameter of about 2(56.3) = miles.

9. GUIDED PRACTICE
for Examples 3, 4 and 5
For the line segment joining the two given points, (a) find the midpoint and (b) write an equation for the perpendicular bisector.
3. (0, 0), (24, 12)
ANSWER
An equation for the perpendicular bisector of AB is
y = x + . 1 3 20
(–2, 6)
4. (–2, 1), (4, –7)
ANSWER
An equation for the perpendicular bisector of AB is
y = x + . 3 4 15
(1 , –7)

10. GUIDED PRACTICE
for Examples 3, 4 and 5
For the line segment joining the two given points, (a) find the midpoint and (b) write an equation for the perpendicular bisector.
5. (3, 8), (–5, –10)
ANSWER
(–1, –1).
An equation for the perpendicular bisector of AB is 4 9 13 9
y =  x  .

11. EXAMPLE 4
GUIDED PRACTICE
for Examples 3, 4 and 5 6.
The points (0, 0), (6, 22), and (16, 8) lie on a circle. Use the method given in Example 5 to find the diameter of the circle.
Q(6, –2) x
A(16, 8) C
B(0, 0)
SOLUTION 20