Name Class Date 8.1Perpendicular Bisectors
of Triangles
Essential Question: H
ow can you use perpendicular bisectors to find the point
that is equidistant from all the vertices of a triangle?
Resource
Locker
Explore Constructing a Circumscribed Circle
Y
A circle that contains all the vertices of a polygon is circumscribed about the polygon.
In the figure, circle C is circumscribed about △XYZ, and circle C is called the circumcircle
of △XYZ. The center of the circumcircle is called the circumcenter of the triangle.
X
In the following activity, you will construct the circumcircle of △PQR. Copy
the triangle onto a separate piece of paper.
A
The circumcircle will pass through P, Q, and R. So, the
center of the circle must be equidistant from all three points.
In particular, the center must be equidistant from Q and R.
The set of points that are equidistant from Q and R is
_
called the of QR​
​  . 
C
Z
Q
C
Use a compass and straightedge to construct the set
R
of points.
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B
The center must also be equidistant from P and R. The set of
points that are equidistant from P and R is called the
_
of PR​
​  . Use a compass and
straightedge to construct the set of points.
P
CThe center must lie at the intersection of the two sets of
points you constructed. Label the point C. Then place the
point of your compass at C and open it to distance CP.
Draw the circumcircle.
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359
Lesson 1
Reflect
1.
Make a Prediction Suppose you started by constructing the set of points equidistant
from P and Q and then constructed the set of points equidistant from Q and R.
Would you have found the same center? Check by doing this construction.
2.
Can you locate the circumcenter of a triangle without using a compass and
straightedge? Explain.
Explain 1
Proving the Concurrency of a Triangle’s
Perpendicular Bisectors
Three or more lines are concurrent if they intersect at the same point. The point of intersection
is called the point of concurrency. You saw in the Explore that the three perpendicular bisectors
of a triangle are concurrent. Now you will prove that the point of concurrency is the circumcenter
of the triangle. That is, the point of concurrency is equidistant from the vertices of the triangle.
Circumcenter Theorem
B
The perpendicular bisectors of the sides of a
triangle intersect at a point that is equidistant
from the vertices of the triangle.
PA = PB = PC
Example 1
P
C
A
_ _
_
Given: Lines ℓ, m, and n are the perpendicular bisectors of AB, BC, and AC,
respectively. P is the intersection of ℓ, m, and n.
A
ℓ
Prove: PA = PB = PC
P is the intersection of ℓ, m, and n. Since P lies on the
_
of AB, PA = PB by the
Theorem. Similarly, P lies on
_
the
of BC, so
= PC. Therefore, PA =
=
by the
Module 8
P
n
C
B
m
Property of Equality.
360
Lesson 1
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Prove the Circumcenter Theorem.
Reflect
3.
Discussion How might you determine whether the circumcenter of a
triangle is always inside the triangle? Make a plan and then determine
whether the circumcenter is always inside the triangle.
Explain 2
Using Properties of Perpendicular Bisectors
You can use the Circumcenter Theorem to find segment lengths in a triangle.
Example 2
_ _
_
KZ, LZ, and MZ are the perpendicular bisectors of △GHJ. Use the
given information to find the length of each segment. Note that the
figure is not drawn to scale.
H
K
G
A
L
Z
J
M
Given: ZM = 7, ZJ = 25, HK = 20
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Find: ZH and HG
Z is the circumcenter of △GHJ, so ZG = ZH = ZJ.
ZJ = 25, so ZH = 25.
_
K is the midpoint of GH, so HG = 2 ⋅ KH = 2 ⋅ 20 = 40.
B
Given: ZH = 85, MZ = 13, HG = 136
Find: KG and ZJ
K is the
Module 8
HG =
of △GHJ, so ZG =
Z is the
ZH =
_
of HG, so KG =
, so ZJ =
=
·
=
.
.
.
361
Lesson 1
Reflect
4.
In △ABC, ∠ACB is a right angle and D is the
circumcenter of the triangle. If CD = 6.5,
what is AB? Explain your reasoning.
C
A
B
D
Your Turn
¯, LZ​
​ KZ​
​  ¯, and MZ​
​ ¯ 
are the perpendicular bisectors of △GHJ. Copy the sketch and label
the given information. Use that information to find the length of each segment. Note
that the figure is not drawn to scale.
H
K
G
Given: ZG = 65, HL = 63, ZL = 16
Find: GK and ZJ
6.
Given: ZM = 25, ZH = 65, GJ = 120
Find: GM and ZG
Module 8
M
L
J
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5.
Z
362
Lesson 1
Explain 3
Finding a Circumcenter on
a Coordinate Plane
Given the vertices of a triangle, you can graph the triangle and use the graph to
find the circumcenter of the triangle.
Example 3
A
Graph the triangle with the given vertices and find the
circumcenter of the triangle.
R( -6, 0 ), S( 0, 4 ), O( 0, 0 )
x = -3
Step 1: Graph the triangle.
y
6
S
Step 2: Find equations for two perpendicular bisectors.
_
Side RO is on the x-axis, so its perpendicular bisector is vertical:
y=2
the line x = -3.
_
Side SO is on the y-axis, so its perpendicular bisector
(-3, 2)
2
x
0 O
-4
R
-2
is horizontal: the line y = 2.
Step 3: Find the intersection of the perpendicular bisectors.
The lines x = -3 and y = 2 intersect at (-3, 2).
(-3, 2) is the circumcenter of △ROS.
B
A(-1, 5), B(5, 5), C(5, -1)
Step 1 Graph the triangle.
7
Step 2 Find equations for two perpendicular bisectors.
_
Side AB is
, so its perpendicular bisector
5
3
is vertical
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y
_
The perpendicular bisector of AB is the line.
.
_
Side BC is
, so the perpendicular bisector of
_
BC is horizontal the line
.
1
-2
0
-2
x
2
4
6
Step 3 Find the intersection of the perpendicular bisectors.
The lines
and
intersect at
.
is the circumcenter of △ABC.
Module 8
363
Lesson 1
Reflect
7.
Draw Conclusions Could a vertex of a triangle also be its circumcenter?
If so, provide an example. If not, explain why not.
Your Turn
Graph the triangle with the given vertices and find the circumcenter of the triangle.
8.
Q​(-4, 0)​, R​(0, 0)​, S​(0, 6)​
6
9.
K​(1, 1)​, L​(1, 7)​, M​(6, 1)​
y
6
4
4
2
2
y
x
-6
-4
-2
0
x
-2
2
-2
0
2
4
6
-2
Elaborate
10. A company that makes and sells bicycles has its largest stores in three cities. The company
wants to build a new factory that is equidistant from each of the stores. Given a map,
how could you identify the location for the new factory?
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11. A sculptor builds a mobile in which a triangle rotates around its circumcenter. Each
vertex traces the shape of a circle as it rotates. What circle does it trace? Explain.
Module 8
364
Lesson 1
12. What If? Suppose you
_are given the vertices of a triangle PQR. You plot the points in a coordinate
plane and notice that PQ is horizontal but neither of the other sides is vertical. How can you identify the
circumcenter of the triangle? Justify your reasoning.
13. Essential Question Check-In How is the point that is equidistant from the three vertices of a triangle
related to the circumcircle of the triangle?
Evaluate: Homework and Practice
• Online Homework
• Hints and Help
• Extra Practice
Construct the circumcircle of each triangle. Label the circumcenter P.
1.
2.
A
A
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B
3.
C
B
4.
B
A
C
A
C
B
C
Module 8
365
Lesson 1
Complete the proof of the Circumcenter Theorem.
¯¯
Use the diagram for Exercise 5–8. ​ ZD,​​ 
 
ZE,​ 
and​  ¯
ZF ​  
are the perpendicular bisectors of △ABC. Use the given
information to find the length of each segment. Note that
the figure is not drawn to scale.
A
D
5.
Given: DZ = 40, ZA = 85, FC = 77
Find: ZC and AC
B
Z
E
Given: FZ = 36, ZA = 85, AB = 150
Find: AD and ZB
7.
Given: AZ = 85, ZE = 51
Find: BC
(Hint: Use the Pythagorean Theorem.)
8.
Analyze Relationships How can you write an algebraic expression for the radius
of the circumcircle of △ABC in Exercises 6–8? Explain.
Module 8
366
C
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6.
F
Lesson 1
Complete the proof of the Circumcenter Theorem.
9.
A
_ _
_
Given: Lines ℓ, m, and n are the perpendicular bisectors of AB​
​  , BC​
​  , and AC​
​  , 
respectively. P is the intersection of ℓ, m, and n.
Prove: PA = PB = PC
ℓ
n
P
C
B
m
Statements
Reasons
1.Lines ℓ, m, and
are the_
perpendicular
_ n_
bisectors of AB​
​  , BC​
​  , and AC​
​  . 
1.
2.P is the intersection of ℓ, m, and n.
2.
3.PA = _
3.P lies on the perpendicular bisector of AB​
​  . 
4. = PC
_
4.P lies on the perpendicular bisector of ​ BC​. 
5.PA = = 5.
_ _
_ _
_
10. ​ PK​, PL​
​  _, and PM​
​   
are the perpendicular bisectors of sides AB​
​  , BC​
​  , 
and AC​
​  . Tell whether the given statement is justified by the figure.
Select the correct answer for each lettered part.
a. AK = KB
b. PA = PB
Not Justified
Justified
Not Justified
Justified
Not Justified
Justified
Not Justified
Justified
Not Justified
B
K
A
M
D
L
C
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c. PM = PL
d. BL = _
​  1 ​ BC
2
e. PK = KD
Justified
P
Module 8
367
Lesson 1
Graph the triangle with the given vertices and find the circumcenter of the triangle.
11. D​(-5, 0)​, E​(0, 0)​, F​(0, 7)​
y
6
4
2
x
-6
-4
-2
0
12. Q​(3, 4)​, R​(7, 4)​, S​(3, -2)​
y
4
2
x
0
2
4
6
-2
13. Represent Real-World Problems For the next Fourth of
July, the towns of Ashton, Bradford, and Clearview will launch
a fireworks display from a boat in the lake. Draw a sketch to
show where the boat should be positioned so that it is the same
distance from all three towns. Justify your sketch.
Ashton
Bradford
Clearview
Final art 3/15/05
F
ge07se_c05l02005a
C
Geometry SE 2007 Texas
Holt Rinehart Winston
Karen Minot
(415)883-6560
H.O.T. Focus on Higher Order Thinking
B
14. Analyze Relationships Explain how can you draw a triangle JKL whose
circumcircle has a radius of 8 centimeters.
Module 8
368
Final art 3/15/05
ge07se_c05l02006a
Geometry SE 2007 Texas
Holt Rinehart Winston
Karen Minot
Lesson 1
© Houghton Mifflin Harcourt Publishing Company
A
_ _
_
15. Persevere in Problem Solving ​ ZD​, ZE​
​   and ZF​
​   are the perpendicular bisectors of
△ABC, which is not drawn to scale.
A
D
B
Z
F
C
E
a. Suppose that ZB = 145, ZD = 100, and ZF = 17. How can you find AB and AC?
b. Find AB and AC.
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c. Can you find BC? If so, explain how and find BC. If not, explain why not.
16. Multiple Representations Given the vertices A​(-2, -2)​,
B​(4, 0)​, and C​(4, 4)​of a triangle, the graph shows how you can use a
graph and construction to locate the circumcenter
_ P of the triangle.
You can draw the perpendicular
bisector
of
​ 
CB​
 
and construct the
_
perpendicular bisector of AB​
​  . Consider how you could identify
P algebraically.
_
passes through its_
midpoint.
a. The perpendicular bisector of AB​
​   
Use the Midpoint Formula to find the midpoint of AB​
​  . 
_
b. What is the slope m of the perpendicular bisector of AB​
​  ? Explain
how you found it.
6
y
C (4, 4)
4
B (4, 0)x
-4
-2
0
-2
A (-2, -2)
-4
2
4
6
_
c. Write an equation of the perpendicular bisector of AB​
​   
and explain how
you can use it find P.
Module 8
369
Lesson 1
Lesson Performance Task
A landscape architect wants to plant a circle of flowers around a triangular garden. She has
sketched the triangle on a coordinate grid with vertices at A​(0, 0)​, B​(8, 12)​, and C​(18, 0)​.
16
y
B (8, 12)
12
8
4
x
A (0, 0)
4
8
12
16 C (18, 0)
Explain how the architect can find the center of the circle that will circumscribe triangle ABC.
Then find the radius of the circumscribed circle.
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Module 8
370
Lesson 1