Name _______________________________________ Date ___________________ Class __________________
Section 5.2: CIRCUMCENTER
Bisectors of Triangles
Perpendicular bisectors
The point of intersection of
MR, MS, and MT is called
MR, MS, and MT are
concurrent because they
intersect at one point.
the circumcenter of NPQ.
Theorem
Example
Circumcenter Theorem
The circumcenter of a triangle is
equidistant from the vertices of
the triangle.
Given: MR, MS, and MT are
the perpendicular bisectors
of NPQ.
Conclusion: MN  MP  MQ
HD, JD , and KD are the perpendicular bisectors of EFG.
Find each length.
1. DG
________________________
3. FJ
________________________
2. EK
________________________
4. DE
________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Geometry
Name _______________________________________ Date __________________ Class __________________
SECTION 5.2: INCENTER
Bisectors of Triangles continued
The point of intersection of
AG, AH, and AJ is called
the incenter of GHJ.
Angle bisectors of GHJ
intersect at one point.
Theorem
Incenter Theorem
The incenter of a triangle is
equidistant from the sides of
the triangle.
Example
Given: AG, AH, and AJ are
the angle bisectors
of GHJ.
Conclusion: AB  AC  AD
PC and PD are angle bisectors of CDE. Find each measure.
7. the distance from P to CE
________________________
8. mPDE
_____________________
KX and KZ are angle bisectors of XYZ. Find each measure.
9. the distance from K to YZ
________________________
10. mKZY
_____________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Geometry
Name _______________________________________ Date __________________ Class __________________
Section 5.2 Practice A
Fill in the blanks to complete each definition or theorem.
1. The circumcenter of a triangle is equidistant from the __________________
of the triangle.
2. When three or more lines __________________ at one point, the lines
are said to be concurrent.
3. The incenter of a triangle is the point where the three __________________
bisectors of a triangle are concurrent.
4. The __________________ of a triangle is equidistant from the sides of the triangle.
5. The __________________ of a triangle is the point where the three
perpendicular bisectors of a triangle are concurrent.
Use the figure for Exercises 6–8. DG, EG, and FG are perpendicular
bisectors ofABC. Find each length.
6. AG _________________
7. DB _________________
8. AF _________________
9. GB _________________
Use the figure for Exercises 10–13. HK and JK are angle bisectors
ofHIJ. Find each measure.
10. the distance from K to JI _________________
11. mHJK _________________
12. mJHK _________________
13. mHJI _________________
Millsville is a town with three large streets that form a triangle.
The town council wants to place a fire station so that it is the
same distance from each of the three streets.
14. Why would the town council want the fire station equidistant from
the large streets?
________________________________________________________________________________________
________________________________________________________________________________________
15. Tell whether the circumcenter or the incenter of the triangle should
be the location of the fire station. ____________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Geometry
Name _______________________________________ Date ___________________ Class __________________
Section 5-3 Centroid
Medians Triangles
AH, BJ, and CG are medians
of a triangle. They each join a
vertex and the midpoint of the
opposite side.
The point of intersection of
the medians is called the
centroid of nABC.
Theorem
Centroid Theorem
The centroid of a triangle is
2
located
of the distance from
3
each vertex to the midpoint of
the opposite side.
Example
Given: AH, CG, and BJ are medians of nABC.
2
2
2
Conclusion: AN  AH, CN  CG, BN  BJ
3
3
3
The centroid is also called the center of gravity because it is the
balance point of the triangle.
In QRS, RX  48 and QW  30. Find each length.
1. RW
________________________
3. QZ
________________________
2. WX
________________________
4. WZ
________________________
In HJK, HD21 and BK18. Find each length.
5. HB
________________________
7. CK
________________________
6. BD
________________________
8. CB
________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Geometry
Name _______________________________________ Date __________________ Class __________________
Altitudes of Triangles - ORTHOCENTER
JD, KE, and LC are altitudes
of a triangle. They are
perpendicular segments that join
a vertex and the line containing
the side opposite the vertex.
The point of intersection of
the altitudes is called the
orthocenter of nJKL.
Medians and Altitudes of Triangles
Fill in the blanks to complete each definition.
1. A median of a triangle is a segment whose endpoints are a vertex of the triangle
and the _____________________ of the opposite side.
2. An altitude of a triangle is a _____________________ segment from a vertex to
the line containing the opposite side.
3. The centroid of a triangle is the point where the three _____________________
are concurrent.
4. The orthocenter of a triangle is the point where the three _____________________
are concurrent.
Use the Centroid Theorem and the figure for Exercises 5–8.
QU , RS , and PT are medians of PQR. RS  21 and VT  5.
Find each length.
5. RV _______________
6. SV _______________
7. TP _______________
8. VP _______________
8. Triangle FGH has coordinates F(3, 1), G(2, 6), and H(4, 1). Find
the centroid.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Geometry
Name _______________________________________ Date __________________ Class __________________
The Triangle Midsegment Theorem
Use the Triangle Midsegment Theorem
to name parts of the figure for
Exercises 1–5.
1. a midsegment ofABC
_____________________
2. a segment parallel to AC
_____________________
3. a segment that has the same length as BD
_____________________
4. a segment that has half the length of AC
_____________________
5. a segment that has twice the length of EC
_____________________
Review Part 2
The Triangle Midsegment Theorem
Use the figure for Exercises 1–6. Find each measure.
1. HI ___________________
2. DF ___________________
3. GE ___________________
4. mHIF ___________________
5. mHGD ___________________ 6. mD ___________________
Review Part 3
Use the Triangle Midsegment Theorem
and the figure for Exercises 14–19.
Find each measure.
14.ST _____________________
15. QR _____________________
16. PU _____________________
17. mSUP _____________________
18. mSUR _____________________
19. mPRQ _____________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Geometry