Name
LESSON
5-2
Date
Class
Reading Strategies
Vocabulary Development
A perpendicular bisector is a line that intersects the side of
a triangle at 90 and passes through its midpoint.
perpendicular
bisector
If three or more lines intersect at the same point, the lines
are concurrent. This may be easy to remember if you think
about three of your favorite television programs being broadcast
concurrently. This means they are on at the same time.
2
The perpendicular bisectors of the sides of a triangle are
concurrent as shown here.
5
6
1. What is a bisector of a triangle?
8
It is a line that divides a side of a triangle
4
3
7
into two congruent parts.
2. What is the perpendicular bisector of RS in RST above?
_
VX
The point where the perpendicular bisectors meet is called
the circumcenter of the triangle and is equidistant from
the vertices of the triangle. It is a point of concurrency
because the three lines intersect at one point.
1
6
4
3. What is the point of concurrency for QRS ?
5
2
point T
7
4. What does equidistant mean?
3
the same distance from two or more objects
or of equal distance
5. In the triangle shown above, ST QT
RT
.
The angle bisectors of a triangle are also concurrent.
Consider the diagram at right:
(
'
In this diagram, point L is the point of concurrency of
the angle bisectors of FHJ. It is called the incenter
of a triangle.
,
&
6. What is the angle bisector of H ?
_
+
)
*
HL
7. What is the angle bisector of F ?
_
FL
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18
Holt Geometry
Name
LESSON
5-2
Date
Class
Name
Reteach
LESSON
Bisectors of Triangles
5-2
continued
The_
point
_of intersection
_
of AG, AH, and AJ is called
the incenter of �GHJ.
�
Theorem
Example
_ _
_
�
Conclusion: AB � AC � AD
WM and WP are angle bisectors of �MNP, and WK � 21.
�
���
_
���
2
_
intersect the same point. Thus the perpendicular bisectors of the sides
of �RST are concurrent.
�
�
���
���
10. m�KZY
52.5°
33
�
��
���
�
15
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Name
Date
�
Class
Holt Geometry
Problem Solving
5-2
Draw a triangle that has the
suburbs as its vertices. Find the
circumcenter of the triangle by
2
2
bisector of each side.
from the sides of the triangle.
Reading Strategies
Vocabulary Development
�
�
�
3. What is the point of concurrency for �QRS?
5. A triangle has vertices Q (–9, 10), R (0, 1),
and S(8, 4). Which is a correct statement
about the incenter and circumcenter of
�QRS ?
F
G
H
J
C F (0, �10), H (8, 0)
D F (0, 10), H(�8, 0)
_
6. RT and TS are perpendicular bisectors
of �ABC. What is the perimeter of �ATC ?
Both points are on �QRS.
Both points are inside �QRS.
Both points are outside �QRS.
One point is inside �QRS, and
one point is outside �QRS.
�
�
or of equal distance
�
56°
F 16�
G 18�
�
17
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All rights reserved.
�
RT
.
�
�
In this diagram, point L is the point of concurrency of
the angle bisectors of �FHJ. It is called the incenter
of a triangle.
�
�
QT
The angle bisectors of a triangle are also concurrent.
Consider the diagram at right:
�
�
_
6. What is the angle bisector of �H ?
�
6
�
the same distance from two or more objects
�
�
�
4. What does equidistant mean?
5. In the triangle shown above, ST �
7. If m�KPN � 44�, find m�JLP.
�
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All rights reserved.
�
�
point T
A F (0, �8), H (10, 0)
�
�
�
�
The point where the perpendicular bisectors meet is called
the circumcenter of the triangle and is equidistant from
the vertices of the triangle. It is a point of concurrency
because the three lines intersect at one point.
�
B F (0, 8), H(�10, 0)
5.2
�
�
_
�
equidistant from X, Y, and Z.
�
perpendicular
bisector
2. What is the perpendicular bisector of RS in �RST above?
�
Choose the best answer.
4. The circumcenter of �FGH is at (4, �5).
If G is at (0, 0), which of the following are
possible coordinates of F and H ?
Holt Geometry
Class
VX
Circumcenter Thm., W is
�
Date
1. What is a bisector of a triangle?
�
of XY , YZ, and ZX. By the
4.5
16
into two congruent parts.
_
D 22.4 units
2
2
It is a line that divides a side of a triangle
Draw the perpendicular bisectors
C 20.9 units
2
2
2
The perpendicular bisectors of the sides of a triangle are
concurrent as shown here.
15 ft; By the Incenter Thm., the
3. A water tower is to be built so that it is the same
distance from the cities at X, Y, and Z. Draw a
sketch on �XYZ to show the location W where
the water tower should be built. Justify your sketch.
B 19.4 units
2
If three or more lines intersect at the same point, the lines
are concurrent. This may be easy to remember if you think
about three of your favorite television programs being broadcast
concurrently. This means they are on at the same time.
incenter of a triangle is equidistant
A 17.2 units
2
A perpendicular bisector is a line that intersects the side of
a triangle at 90� and passes through its midpoint.
drawing the perpendicular
_ _
2
2
2
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
LESSON
2. A fountain is in a triangular sitting area
of a mall, �ABC. A diagram shows that
the fountain is at the point where the
angle bisectors of �ABC are concurrent.
If the distance from the fountain to one
wall is 15 feet, what is the distance from
the fountain to another wall? Explain.
2
2
Name
Bisectors of Triangles
1. A new dog park is being planned.
Describe how to find a location for
the park so that it is the same distance
from three suburbs.
_
2
(RZ ) : _c_
2
�
KX and KZ are angle bisectors of �XYZ. Find each measure.
5-2
� b � ac ;
�
� � � � a___________
2b
� b � ac ;
(SZ ) : � a � _c_ � � � b � a____________
�
2b
2
�
b
�
ac
c
a
_
_
____________
(TZ ) : � c � � � � �
�
2b
2
�
_
LESSON
2. Let point Z be the point of concurrency of the three perpendicular bisectors of the
sides of �RST above. Follow these steps to prove that point Z is equidistant from
the vertices of �RST. In other words, prove that RZ � SZ � TZ. Use the Distance
2
2
2
Formula to write expressions for (RZ ) , (SZ ) , and (TZ ) .
�
34°
_
_
point as the perpendicular bisectors of ST and RT, all three lines
8. m�PDE
9. the distance from K to YZ
_
_
���
�
2
Since the perpendicular bisectors of RS and RT intersect in the same
PC and PD are angle bisectors of �CDE. Find each measure.
_
2
d. Use the results of parts b and c to complete the proof.
Subtract 136° from each side.
9
�
� c2, a � b2b � ac �
�
7. the distance from P to CE
��
�
__ ____________
_
_
_
c. Use a system of equations_
to find the
_ coordinates of the point where the
perpendicular bisectors of ST and RT intersect.
�
�
m�WPN � _1_ m�NPM
Def. of � bisector
2
m�WPN � _1_ (44°) � 22° Substitute.
2
_
_
The distance from W to MN and NP is 21 by the Incenter Theorem.
_
�
__ ____________
Substitute.
m�NPM � 44°
� ������
_
� c2, a � b2b � ac �
��
�
� Sum Thm.
�
� � ������
b. Use a system of equations_
to find the
_coordinates of the point where the
perpendicular bisectors of RS and RT intersect.
�
m�NMP � 2(32°) � 64° Substitute.
64° � 72° � m�NPM � 180°
� ��
2
Def. of � bisector
m�NMP � m�N � m�NPM � 180°
_
�
�
Find m�WPN and the distance from W to MN and NP .
m�NMP � 2m�NMW
� ������
�c x�a
�c ;
______
______
RS : y � _b_ � � _a_ x � _a_ ; ST : y � _b_ � � a
b
b
2
2
2
2
_
c
RT : x � __
2
�
_
_
�
a. Write equations for the perpendicular bisectors of RS, ST , and RT. (Hint: If a
line having slope m passes through a point P(x1, y1), then an equation of the
line is y � y1 � m(x � x1).)
�
�
�
_
The Perpendicular Bisectors of the Sides of a Triangle
_ _
Given: AG, AH, and AJ are
the angle bisectors
of �GHJ.
Incenter Theorem
The incenter of a triangle is
equidistant from the sides of
the triangle.
Challenge
1. Refer to the figure at right. Follow these steps to prove that
the perpendicular bisectors of the sides of �RST are
concurrent. That is, you will prove that all three
perpendicular bisectors intersect at a single point.
�
�
Class
In Lesson 5-2, you investigated the three perpendicular bisectors
of the sides of a triangle and discovered some surprising facts
about them. On this page, you will see how coordinate methods
can help you prove those facts.
�
Angle bisectors of �GHJ
intersect at one point.
Date
�
�
�
HL
�
_
7. What is the angle bisector of �F ?
H 23�
J 32�
FL
Holt Geometry
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All rights reserved.
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18
Holt Geometry
Holt Geometry