Page 2 of 5
Chapter Review
CHAPTER
5
• perpendicular bisector,
• equidistant from two lines,
• circumcenter of a triangle,
• equidistant from two points,
• perpendicular bisector of a
• angle bisector of a triangle,
• centroid of a triangle, p. 279 • altitude of a triangle, p. 281 • orthocenter of a triangle,
• distance from a point to a
• concurrent lines, p. 272 • point of concurrency, p. 272
• incenter of a triangle, p. 274 • median of a triangle, p. 279
• midsegment of a triangle, p. 287
p. 264
p. 266
p. 264
line, p. 266
triangle, p. 272
p. 273 p. 274
p. 281
• indirect proof, p. 302
5.1
Examples on pp. 264–267
PERPENDICULARS AND BISECTORS Æ˘
In the figure, AD is the angle bisector of Æ ™BAC and the perpendicular bisector of BC. You know that BE = CE by the definition of perpendicular bisector and that AB = AC by the Perpendicular Bisector Theorem. Because Æ˘ Æ˘ Æ Æ DP fi AP and DQ fi AQ , then DP and DQ are the distances from D to the sides of ™PAQ and you know that DP = DQ by the Angle Bisector Theorem. EXAMPLES
D
q
E
P B
C A
In Exercises 1–3, use the diagram. Æ ˘
Æ
1. If SQ is the perpendicular bisector of RT, explain how you know that Æ
Æ
Æ
R
Æ
RQ £ TQ and RS £ TS . Æ
Æ
U
2. If UR £ UT, what can you conclude about U? Æ˘
S
Æ˘
3. If Q is equidistant from SR and ST , what can you conclude about Q?
5.2
q T
Examples on pp. 272–274
BISECTORS OF A TRIANGLE The perpendicular bisectors of a triangle intersect at the circumcenter, which is equidistant from the vertices of the triangle. The angle bisectors of a triangle intersect at the incenter, which is equidistant from the sides of the triangle. EXAMPLES
4. The perpendicular bisectors of ¤RST
intersect at K. Find KR.
R
K 12
5. The angle bisectors of ¤XYZ intersect at W.
Find WB. S
Z
32
A 8
T
310
Chapter 5 Properties of Triangles
Y
B
W 10
X
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5.3
Examples on pp. 279–281
MEDIANS AND ALTITUDES OF A TRIANGLE EXAMPLES The medians of a triangle intersect at the centroid. The lines containing the altitudes of a triangle intersect at the orthocenter. ¯˘ ¯˘ ¯˘ HN , JM , and KL intersect at Q.
B
2 3
AP = }}AD
F
D
H N J
q
M
P L A
E
K
C
Name the special segments and point of concurrency of the triangle. 6.
7.
7
8.
6
7
9.
6 8
8
¤XYZ has vertices X(0, 0), Y(º4, 0), and Z(0, 6). Find the coordinates of the indicated point. 10. the centroid of ¤XYZ
5.4
11. the orthocenter of ¤XYZ Examples on pp. 287–289
MIDSEGMENT THEOREM A midsegment of a triangle connects the midpoints of two sides of the triangle. By the Midsegment Theorem, a midsegment of a triangle is parallel to the third side and its length is half the length of the third side. EXAMPLES
Æ
Æ
1 2
DE ∞ AB , DE = }}AB
C E D
5
B 10
A
In Exercises 12 and 13, the midpoints of the sides of ¤HJK are L(4, 3), M(8, 3), and N(6, 1). 12. Find the coordinates of the vertices of the triangle. 13. Show that each midsegment is parallel to a side of the triangle. 14. Find the perimeter of ¤BCD.
15. Find the perimeter of ¤STU.
B T
G
E 12 D
R
F 22
10
9 C
U
P
S 24
9 q
Chapter Review
311
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5.5
Examples on pp. 295–297
INEQUALITIES IN ONE TRIANGLE In a triangle, the side and the angle of greatest measurement are always opposite each other. In the diagram, Æ the largest angle, ™MNQ, is opposite the longest side, MQ. EXAMPLES
M
41.48
5
By the Exterior Angle Inequality, m™MQP > m™N and m™MQP > m™M. By the Triangle Inequality, MN + NQ > MQ, NQ + MQ > MN, and MN + MQ > NQ.
6
55.88 124.28
82.88 4
N
P
q
In Exercises 16–19, write the angle and side measurements in order from least to greatest. 16.
17.
C
25 10
20.
508
19.
H
K
23
558
9
708 35
D A
18. J
F
E G
L
M
B
8
FENCING A GARDEN You are enclosing a triangular garden region with a
fence. You have measured two sides of the garden to be 100 feet and 200 feet. What is the maximum length of fencing you need? Explain.
5.6
Examples on pp. 302–304
INDIRECT PROOF AND INEQUALITIES IN TWO TRIANGLES EXAMPLES
Æ
Æ
Æ
Æ
AB £ DE and BC £ EF
E
F
Hinge Theorem: If m™E > m™B, then DF > AC.
B
C D
Converse of the Hinge Theorem: If DF > AC, then m™E > m™B.
A
In Exercises 21–23, complete with <, >, or =.
? CB 21. AB ooo
? m™2 22. m™1 ooo C
R
? VS 23. TU ooo
16
S
1
1268
D 928 888 A
U
T
2 B
P
15
S
q
24. Write the first statement for an indirect proof of this situation: In a ¤MPQ, if
™M £ ™Q, then ¤MPQ is isosceles. 25. Write an indirect proof to show that no triangle has two right angles. 312
W
1268
Chapter 5 Properties of Triangles
V
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CHAPTER
5
Chapter Test
In Exercises 1–5, complete the statement with the word always, sometimes, or never. 1. If P is the circumcenter of ¤RST, then PR, PS, and PT are oooooo o ? equal. Æ˘
Æ
Æ
2. If BD bisects ™ABC, then AD and CD are oooooo o ? congruent. 3. The incenter of a triangle oooooo o ? lies outside the triangle. 4. The length of a median of a triangle is oooooo o ? equal to the length of a midsegment. Æ
Æ
Æ
Æ
5. If AM is the altitude to side BC of ¤ABC, then AM is oooooo o ? shorter than AB. In Exercises 6–10, use the diagram.
C
6. Find each length. a. HC
b. HB
c. HE
d. BC
E
F
7. Point H is the oooooo o ? of the triangle.
9.9
H 6
Æ
8. CG is a(n) oooooo o ? , oooooo o ? , and oooooo o ? of ¤ABC . o ? , oooooo
A
Æ
9. EF = oooooo o ? Theorem. o ? and EF ∞ oooooo o ? by the oooooo
G
8
B
10. Compare the measures of ™ACB and ™BAC. Justify your answer. 11.
LANDSCAPE DESIGN You are designing a circular swimming pool for a triangular lawn surrounded by apartment buildings. You want the center of the pool to be equidistant from the three sidewalks. Explain how you can locate the center of the pool.
In Exercises 12–14, use the photo of the three-legged tripod. 12. As the legs of a tripod are spread apart, which theorem
guarantees that the angles between each pair of legs get larger? 13. Each leg of a tripod can extend to a length of 5 feet. What is
the maximum possible distance between the ends of two legs? Æ Æ
Æ
14. Let OA, OB, and OC represent the legs of a tripod. Draw
and label a sketch. Suppose the legs are congruent and Æ Æ m™AOC > m™BOC. Compare the lengths of AC and BC. In Exercises 15 and 16, use the diagram at the right. 15. Write a two-column proof. B
A
GIVEN c AC = BC PROVE c BE < AE
C
16. Write an indirect proof. GIVEN c AD ≠ AB PROVE c m™D ≠ m™ABC
E
D
Chapter Test
313
