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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
2
)=(
– 5 + (– 1)
2
,
1+6
2
)=(
7
– 3,
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
)=(
–3 + 5
2
,
4 + 6
2
) = (1, 5)
EXAMPLE 4
Find a perpendicular bisector
STEP 2
Calculate the slope of AB
y2 – y1
6 – 4
m = x –x =
= 2 = 1
5 – (– 3)
4
8
2
1
STEP 3
Find the slope of the perpendicular bisector:
– 1 = – 1 =–4
m
1/4
EXAMPLE 4
Find a perpendicular bisector
STEP 4
Use point-slope form:
y – 5 = – 4(x – 1), or y = – 4x + 9.
ANSWER
An equation for the perpendicular bisector of AB is
y = – 4x + 9.
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.)
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 + 110
Perpendicular bisector of OB
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
3x + 110 = – x + 34
Write first equation.
Substitute for y.
Simplify.
4x = – 76
x = – 19
Solve for x.
y = – (– 19) + 34 Substitute the x-value into the first
equation.
Solve for y.
y = 53
The center of the circle is C (– 19, 53).
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.
OC =
(–19 – 0)2 + (53 – 0)2 =
3170
56.3
Use (x1, y1) = (0, 0) and (x2, y2) = (–19, 53).
ANSWER
The crater has a diameter of about 2(56.3) = 112.6 miles.
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)
SOLUTION
Let (x1, y1 ) = (0, 0) and ( x2, y2 ) = (– 4, 12).
(
x1 + x2 , y1 + y2
2
2
)= (
0 + (– 4)
2
,
0 + 12
2
)
=
(
4 12
– , 2
2
= (–2, 6)
)
EXAMPLE
4
GUIDED PRACTICE
for Examples 3, 4 and 5
SOLUTION
STEP 1
Find the midpoint of the line segment.
(
x1 + x2 , y1 + y2
2
2
)=(
0 + (–4)
2
,
0+2
2
STEP 2
Calculate the slope
y2 – y1
12 – 0
12 = –3
m = x –x =
=
–4
–4
–
0
2
1
) = (–2, 6)
EXAMPLE
4
GUIDED PRACTICE
for Examples 3, 4 and 5
STEP 3
Find the slope of the perpendicular bisector:
–
1
=
m
–1
1
–3 = 3
STEP 4
Use point-slope form:
1
20
y  6 = 1 (x + 2),
or y = 3 x + 3.
3
ANSWER
An equation for the perpendicular bisector of AB is
1
20
y = 3x + .
3
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.
4.
(–2, 1), (4, –7)
SOLUTION
Let (x1, y1 ) = (–2, 1) and ( x2, y2 ) = (4, – 7).
(
x1 + x2 , y1 + y2
2
2
)= (
midpoint is ( 1, –3)
–2+4
2
,
1 + (–7)
2
)
= (1 , –7)
EXAMPLE
4
GUIDED PRACTICE
for Examples 3, 4 and 5
SOLUTION
STEP 1
Find the midpoint of the line segment.
(
x1 + x2 , y1 + y2
2
2
)=(
–2+4
2
,
1 + (–7)
2
STEP 2
Calculate the slope
y2 – y1
–7 – 1
–8
m = x –x =
=
6
4 – (–2)
2
1
= 4
3
)
= (1 , –7)
EXAMPLE
4
GUIDED PRACTICE
for Examples 3, 4 and 5
STEP 3
Find the slope of the perpendicular bisector:
–
1
=
m
–1
3
= 4
4
3
STEP 4
Use point-slope form:
3
15
y + 7 = 3 (x  1),
or y = 4 x + 4.
4
ANSWER
An equation for the perpendicular bisector of AB is
3
15
y = 4x + .
4
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)
SOLUTION
Let (x1, y1 ) = (3, 8) and ( x2, y2 ) = (– 5, –10).
(
x1 + x2 , y1 + y2
2
2
)= (
midpoint is (–1 , –1)
3 + (– 5)
2
,
8 + (–10)
2
) = (–1 , –1)
EXAMPLE
4
GUIDED PRACTICE
for Examples 3, 4 and 5
SOLUTION
STEP 1
Find the midpoint of the line segment.
(
x1 + x2 , y1 + y2
2
2
)= (
3 + (– 5)
2
,
8 + (–10)
= (–1 , –1)
2
STEP 2
Calculate the slope
y2 – y1
–10 – 8
–18
9
m = x –x =
=
=
–
8
4
–5
–
3
2
1
)
EXAMPLE
4
GUIDED PRACTICE
for Examples 3, 4 and 5
STEP 3
Find the slope of the perpendicular bisector:
–
1
=
m
–1
9
4
4
= 9
STEP 4
Use point-slope form:
y  1 =  4 (x  1),
or y =  4 x  13.
9
9
9
ANSWER
An equation for the perpendicular bisector of AB is
y =  4 x  13.
9
9
EXAMPLE
4
GUIDED PRACTICE
6.
for Examples 3, 4 and 5
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.
C
B(0, 0)
A(16, 8)
Q(6, –2)
SOLUTION
20
x