CH6.1-6.3 Review Worksheet

Name _______________________________________ Date ___________________ Class __________________
Section 6.1
Medians of 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 ∆ABC.
Main Ideas from class:
1.
Short piece x 2 = long piece.
2.
longpiece
= shortpiece
2
3.
median
= shortpiece
3
In ∆ABC above, suppose AH = 18, BN = 10, and AJ = 11, find the following:
AN = ______
NJ = ______
JC = ______
NH = ______
JB = ______
AC = = ______
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
________________________
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Holt McDougal Geometry
Name _______________________________________ Date __________________ Class __________________
Section 6.1
Medians of Triangles
Fill in the blanks to complete each definition.
9. A median of a triangle is a segment whose endpoints are a vertex of the triangle
and the _____________________ of the opposite side.
10. The centroid of a triangle is the point where the three _____________________
are concurrent.
The centroid is also called the center of gravity because it is the balance point of the
triangle.
11. By holding a tray at the center of gravity, a waiter can carry with one hand a large triangular
tray loaded with several dishes.If the vertices of the tray have coordinates A(0, 0), B(9, 0),
and C(0, 6), find the coordinates of the balance point (centroid) of the tray. (Hint: The xcoordinate of the centroid is the average of the x-coordinates of the three vertices, and
the y-coordinate of the centroid is the average of the y-coordinates of the three vertices.)
12.The diagram shows the coordinates of the
vertices of a triangular patio umbrella. The
umbrella will rest on a pole that will support it.
Where should the pole be attached so that the
umbrella is balanced?
13. In a plan for a triangular wind chime, the coordinates of the vertices are J(10, 2),
K(7, 6), and L(12, 10). At what coordinates should the manufacturer attach the chain the
from which it will hang in order for the chime to be balanced?
14. Triangle PQR has vertices at P(−3, 5), Q(−1, 7), and R(3, 1). Find the coordinates of the
centroid.
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Holt McDougal Geometry
Name _______________________________________ Date __________________ Class __________________
Section 6.2
Altitudes of Triangles
altitude
An altitude of a triangle is a perpendicular
segment from a vertex to the line containing
the opposite side.
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 ∆JKL.
Find the orthocenter of ∆ABC
with vertices A(–3, 3), B(3, 7), and C(3, 0).
∆
Triangle FGH has coordinates F(−
−3, 1), G(2, 6), and H(4, 1).
1. Find the orthocenter using the above process.
2. An altitude of a triangle is a _____________________ segment from a vertex to
the line containing the opposite side.
3. The orthocenter of a triangle is the point where the three _____________________
are concurrent.
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Holt McDougal Geometry
Name _______________________________________ Date __________________ Class __________________
Section 6.2
Perpendicular Bisectors
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.
Perpendicular bisectors
The point of intersection of
MR, MS, and MT are
concurrent because they
intersect at one point.
MR, MS, and MT is called
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
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 __________________ of a triangle is the point where the three
perpendicular bisectors of a triangle are concurrent.
Use the figure for Exercises 1 and 2. SV, TV, and UV are
perpendicular bisectors of the sides of ∆PQR. Find each length.
4.
RV ________________
5.
TR ________________
HD, JD , and KD are the perpendicular bisectors of ∆EFG.
Find each length.
6. DG = __________
7. EK= __________
8. FJ= __________
9. DE= __________
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Holt McDougal Geometry
Name _______________________________________ Date __________________ Class __________________
Section 6.3
Bisectors of Triangles
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
Class example:
WM and WP are angle bisectors of ∆MNP, and WK = 21.
Find m∠WMP and m∠WPN
The distance from W to MN =
PC and PD are angle bisectors of ∆CDE. Find each measure.
1. the distance from P to CE
________________________
2. m∠PDE
_____________________
KX and KZ are angle bisectors of ∆XYZ. Find each measure.
3. the distance from K to YZ
________________________
4. m∠KZY
_____________________
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Holt McDougal Geometry
Name _______________________________________ Date __________________ Class __________________
5 . The incenter of a triangle is the point where the three __________________
bisectors of a triangle are concurrent.
6. The __________________ of a triangle is equidistant from the sides of the triangle.
Use the figure for Exercises 7 and 8. GJ and IJ are angle
bisectors of ∆GHI. Find each measure.
7. the distance from J to GH _________________
8. m∠JGK _________________
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.
9. Why would the town council want the fire station equidistant from
the large streets?
________________________________________________________________________________________
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 _________________
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Holt McDougal Geometry

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