Copyright © by Holt, Rinehart and Winston. 172 GeometryAll rights reserved.
Vocabulary8
CHAPTER
The table contains important vocabulary terms from Chapter 8. As you workthrough the chapter, fill in the page number, definition, and a clarifying example.
angle of depression
angle of elevation
componentform
cosine
equal vectors
geometric mean
Term Page Definition Clarifying Example
Copyright © by Holt, Rinehart and Winston. 172 GeometryAll rights reserved.
Vocabulary8
CHAPTER
The table contains important vocabulary terms from Chapter 8. As you workthrough the chapter, fill in the page number, definition, and a clarifying example.
angle of depression
angle of elevation
componentform
cosine
equal vectors
geometric mean
544
544
559
525
561
519
The angle formed by ahorizontal line and aline of sight to a pointbelow.
The angle formed by ahorizontal line and aline of sight to a pointabove.
The form of a vectorthat lists the verticaland horizontal changefrom the initial point tothe terminal point.
In a right triangle, thecosine of angle A is theratio of the length ofthe side adjacent toangle A to the length ofthe hypotenuse. It isthe reciprocal of thesecant function.
Two vectors that havethe same magnitudeand the same direction.
For positive numbers aand b, the positivenumber x such that �ax� � �b
x�. In a geometric
sequence, a term thatcomes between twogiven nonconsecutiveterms of the sequence.
The component form ofCD�� is �2.3�.
Term Page Definition Clarifying Example
Angle ofdepression
Angle ofelevation
adjacentleg
hypotenuse
A
C 2
3
D
Given: 4 and 9
Let x be the geometricmean.x2 � (4)(9) � 36x � 6
Copyright © by Holt, Rinehart and Winston. 173 GeometryAll rights reserved.
CHAPTER 8 VOCABULARY CONTINUED
magnitude
parallel vectors
resultant vector
sine
tangent of anangle
trigonometricratio
vector
Term Page Definition Clarifying Example
Copyright © by Holt, Rinehart and Winston. 173 GeometryAll rights reserved.
CHAPTER 8 VOCABULARY CONTINUED
magnitude
parallel vectors
resultant vector
sine
tangent of anangle
trigonometricratio
vector
560
561
561
525
525
525
559
The length of a vector,written �AB�� � or �v�� � .
Vectors with the sameor opposite direction.
The vector thatrepresents the sum oftwo given vectors.
In a right triangle, theratio of the length ofthe side opposite �A tothe length of thehypotenuse.
In a right triangle, theratio of the length ofthe side opposite �A tothe length of the legadjacent to �A.
A ratio of two sides of aright triangle.
A quantity that has bothmagnitude anddirection.
The sine of an angle isthe ratio of the lengthof the leg opposite theangle to the length ofthe hypotenuse.
Term Page Definition Clarifying Example
� �4, �2��� �(4 � 0�)2 � (��2 � 0�)2�� �20� 4.5
parallel vectors
u
resultantvector
v
u + v
�
� �
�
oppositeleg
hypotenuse
A
oppositeside
Aadjacent leg
4
–2
2y
x0
(4, –2)
C 2
3
D
Chapter Review8
CHAPTER
Copyright © by Holt, Rinehart and Winston. 184 GeometryAll rights reserved.
8-1 Similarity in Right Triangles
Find the geometric mean of each pair of numbers. If necessary, give the answers in simplest radical form.
1. 10 and 12 2. 4.5 and 32 3. �54� and �
190�
Find x, y, and z.
4. 5. 6.
7. A surveyor needs to find the distance across a pond.What is AD to the nearest tenth of a meter?
8-2 Trigonometric Ratios
Use a special right triangle to write each trigonometric ratio as a fraction.
8. tan 60° 9. cos 60° 10. sin 60°
Use your calculator to find each trigonometric ratio. Round to the nearesthundredth.
11. sin 12° 12. cos 59° 13. tan 17°
B
AD
C
42 m
15 m
y z12
24 x
z
x
y36y
2 8
x z
18�5�
Chapter Review8
CHAPTER
Copyright © by Holt, Rinehart and Winston. 184 GeometryAll rights reserved.
8-1 Similarity in Right Triangles
Find the geometric mean of each pair of numbers. If necessary, give the answers in simplest radical form.
1. 10 and 12 2. 4.5 and 32 3. �54� and �
190�
Find x, y, and z.
4. 5. 6.
7. A surveyor needs to find the distance across a pond.What is AD to the nearest tenth of a meter?
8-2 Trigonometric Ratios
Use a special right triangle to write each trigonometric ratio as a fraction.
8. tan 60° 9. cos 60° 10. sin 60°
Use your calculator to find each trigonometric ratio. Round to the nearesthundredth.
11. sin 12° 12. cos 59° 13. tan 17°
0.310.520.21
�12��3�
48.9 m
B
AD
C
42 m
15 m
x � 6, y � 12�5�, z � 6�5�x � 9, y � 18, z � 9�5�x � 4; y � 2�5�; z � 4�5�
y z12
24 x
z
x
y36y
2 8
x z
�5�
62�
�122�30�
�3��
2
18�5�
Copyright © by Holt, Rinehart and Winston. 185 GeometryAll rights reserved.
Find each length. Round to the nearest hundredth.
14. LK 15. AB 16. WX
8-3 Solving Right Triangles
Find the unknown measures. Round lengths to the nearest hundredthand angle measures to the nearest degree.
17. 18. 19.
20. A bike trail has a sign warning bikers of a 7° inclinefor the next 12 miles. To the nearest tenth of a mile, find the vertical distance of the incline, h.
8-4 Angles of Elevation and Depression
21. From the top of a lighthouse to a ship the angle of depression is 28°. The top of the lighthouse is located 175 feet above the water. What is the horizontal distance from the lighthouse to the ship? Round to the nearest foot.
28
175 ft
7°
12 mih
32°
9.5
D E
F
3.6
5.5 ML
N
25° BC
A
14
51°xw
y
9.6
40°C
A
B
10
35°
J
L K
20
CHAPTER 8 REVIEW CONTINUED
Copyright © by Holt, Rinehart and Winston. 185 GeometryAll rights reserved.
Find each length. Round to the nearest hundredth.
14. LK 15. AB 16. WX
8-3 Solving Right Triangles
Find the unknown measures. Round lengths to the nearest hundredthand angle measures to the nearest degree.
17. 18. 19.
20. A bike trail has a sign warning bikers of a 7° inclinefor the next 12 miles. To the nearest tenth of a mile, find the vertical distance of the incline, h.
8-4 Angles of Elevation and Depression
21. From the top of a lighthouse to a ship the angle of depression is 28°. The top of the lighthouse is located 175 feet above the water. What is the horizontal distance from the lighthouse to the ship? Round to the nearest foot.
329 feet
28
175 ft
7°
12 mih
1.5 mi
�F � 58˚; DE � 8.06;DF � 5.03
�L � 33˚; �N � 57;LN � 6.57
�A � 65˚; AB � 33.13;CB � 30.02
32°
9.5
D E
F
3.6
5.5 ML
N
25° BC
A
14
6.046.4328.56
51°xw
y
9.6
40°C
A
B
10
35°
J
L K
20
CHAPTER 8 REVIEW CONTINUED
Copyright © by Holt, Rinehart and Winston. 186 GeometryAll rights reserved.
22. The angle of elevation of the sun is 82°, a water tower casts a shadow that is 22 m long. What is the height of the water tower to the nearest tenth of a meter?
8-5 Law of Sines and Law of Cosines
Find each measure. Round lengths to the nearest tenth and anglemeasures to the nearest degree.
23. m�P 24. AB 25. AC
26. JL 27. m�A 28. RP
29°
12
15
P
R Q16
19 12
A C
B
59°
13
9
JK
L
34°71°
115
C
AB
103°24°
73C
A
B42°
43
27P
Q
R
82
CHAPTER 8 REVIEW CONTINUED
Copyright © by Holt, Rinehart and Winston. 186 GeometryAll rights reserved.
22. The angle of elevation of the sun is 82°, a water tower casts a shadow that is 22 m long. What is the height of the water tower to the nearest tenth of a meter?
8-5 Law of Sines and Law of Cosines
Find each measure. Round lengths to the nearest tenth and anglemeasures to the nearest degree.
23. m�P 24. AB 25. AC
26. JL 27. m�A 28. RP
7.439˚11.4
29°
12
15
P
R Q16
19 12
A C
B
59°
13
9
JK
L
66.659.825˚
34°71°
115
C
AB
103°24°
73C
A
B42°
43
27P
Q
R
156.5 m
82
CHAPTER 8 REVIEW CONTINUED
Copyright © by Holt, Rinehart and Winston. 187 GeometryAll rights reserved.
8-6 Vectors
Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth.
29. �4, 1� 30. ��3, �5� 31. �0, 6�
Draw each vector on a coordinate plane. Find the direction of the vector to thenearest degree.
32. A wind velocity is given by 33. The path of a hiker is given by the vector �2, 4�. the vector �7, 4�.
34. To reach an island, a ship leaves port and sails for 10 km at a bearing of N 48° E. It then sails due east for 14 km.What are the magnitude and direction of the voyage directly from the port to the island? Round the distance to the nearest tenth of a kilometer and the direction to the nearest degree.
48
14 km Island10 km
W E E E
S
N
2
–2
4
6
y
62 4x
y
x2 4–2
2
–2
4
6
–4
y
x2 4–2
2
–2
4
6
–4
2 4–2
2
–2
4
y
x
–4
–42 4–2
2
–2
4
y
x
–4
–4
CHAPTER 8 REVIEW CONTINUED
Copyright © by Holt, Rinehart and Winston. 187 GeometryAll rights reserved.
8-6 Vectors
Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth.
29. �4, 1� 30. ��3, �5� 31. �0, 6�
Draw each vector on a coordinate plane. Find the direction of the vector to thenearest degree.
32. A wind velocity is given by 33. The path of a hiker is given by the vector �2, 4�. the vector �7, 4�.
34. To reach an island, a ship leaves port and sails for 10 km at a bearing of N 48° E. It then sails due east for 14 km.What are the magnitude and direction of the voyage directly from the port to the island? Round the distance to the nearest tenth of a kilometer and the direction to the nearest degree.
22.5 km; N73 E
48
14 km Island10 km
W E E E
S
N
30°
2
–2
4
6
y
62 4x
63°y
x2 4–2
2
–2
4
6
–4
6�34� 5.8�17� 4.1
y
x2 4–2
2
–2
4
6
–4
2 4–2
2
–2
4
y
x
–4
–42 4–2
2
–2
4
y
x
–4
–4
CHAPTER 8 REVIEW CONTINUED
Copyright © by Holt, Rinehart and Winston. 188 GeometryAll rights reserved.
Theorem 8-1-1 The altitude to the hypotenuse of a right triangle forms twotriangles that are similar to each other and to the originaltriangle.
Corollary 8-1-2 The length of the altitude to the hypotenuse of a righttriangle is the geometric mean of the lengths of the twosegments of the hypotenuse.
Corollary 8-1-3 The length of a leg of a right triangle is the geometric meanof the lengths of the hypotenuse and the segments of thehypotenuse adjacent to that leg.
Theorem 8-5-1 (The Law of Sines) For any �ABC with side lengths a, b,and c,
�sin
aA
� � �sin
bB
� � �sin
cC
�.
Theorem 8-5-2 (The Law of Cosines) For any �ABC with side lengths a, b,and c,
a2 � b2 � c2 � 2bc cos A,
b2 � a2 � c2 � 2ac cos B, and
c2 � a2 � b2 � 2ab cos C.
Postulates and Theorems8
CHAPTER
Copyright © by Holt, Rinehart and Winston. 188 GeometryAll rights reserved.
Theorem 8-1-1 The altitude to the hypotenuse of a right triangle forms twotriangles that are similar to each other and to the originaltriangle.
Corollary 8-1-2 The length of the altitude to the hypotenuse of a righttriangle is the geometric mean of the lengths of the twosegments of the hypotenuse.
Corollary 8-1-3 The length of a leg of a right triangle is the geometric meanof the lengths of the hypotenuse and the segments of thehypotenuse adjacent to that leg.
Theorem 8-5-1 (The Law of Sines) For any �ABC with side lengths a, b,and c,
�sin
aA
� � �sin
bB
� � �sin
cC
�.
Theorem 8-5-2 (The Law of Cosines) For any �ABC with side lengths a, b,and c,
a2 � b2 � c2 � 2bc cos A,
b2 � a2 � c2 � 2ac cos B, and
c2 � a2 � b2 � 2ab cos C.
Postulates and Theorems8
CHAPTER
Big Ideas8
CHAPTER
Copyright © by Holt, Rinehart and Winston. 189 GeometryAll rights reserved.
Answer these questions to summarize the important concepts fromChapter 8 in your own words.
1. Explain the trigonometric ratios.
2. Explain what vectors are and how they are used.
3. Explain what solving a right triangle means.
4. Explain why and how the Law of Sines and Law of Cosines are used.
For more review of Chapter 8:
• Complete the Chapter 8 Study Guide and Review on pages 572–575 ofyour textbook.
• Complete the Ready to Go On quizzes on pages 543 and 569 of yourtextbook.
Big Ideas8
CHAPTER
Copyright © by Holt, Rinehart and Winston. 189 GeometryAll rights reserved.
Answer these questions to summarize the important concepts fromChapter 8 in your own words.
1. Explain the trigonometric ratios.
2. Explain what vectors are and how they are used.
3. Explain what solving a right triangle means.
4. Explain why and how the Law of Sines and Law of Cosines are used.
For more review of Chapter 8:
• Complete the Chapter 8 Study Guide and Review on pages 572–575 ofyour textbook.
• Complete the Ready to Go On quizzes on pages 543 and 569 of yourtextbook.
Answers will vary. Possible answer: The Law of Sines and Cosinesare used to solve oblique triangles, triangles that have no rightangles. The Law of Sines is best used when given one side and twoangles. The Law of Cosines is best used when given three sides ortwo sides and an included angle.
Answers will vary. Possible answer: Solving a right triangle meansto find the measures of all angles and all sides.
Answers will vary. Possible answer: A vector can represent speedand direction. It has both length and direction. It has a horizontaland vertical change from its initial point to its terminal point.
Answers will vary. Possible answer: The trigonometric ratios aresine, cosine and tangent. They are used to solve right triangles.Sine is the opposite side over the hypotenuse. Cosine is theadjacent side over the hypotenuse. Tangent is the opposite sideover the adjacent side.
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