Geometry

CHAPTER

Lesson 7-1 Get Ready! 8

Solving Proportions

Algebra  Solve for x. If necessary, round answers to the nearest thousandth.

1. 0.2734 = 1x7 2. 0.5858 = 24 3. 0.8572 = 5271 4. 0.5 = 3x x 5
x x +

Lesson 7-3 Proving Triangles Similar

Name the postulate or theorem that proves each pair of triangles similar.

5. CD } AB 6. 21 7. JK # ML
CD
E 9.6 27 J
14 5N
AB 6.4 18 15

M 21 K 28 L

Lesson 7-4 Similarity in Right Triangles

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x 9 15

B 16 D 9 A 32 D B x D 16 A
4
D C
Cx Ax

16 B hsm11gmse_08co_t08583.ai

hsm11gmse_08co_t08579.ai

Looking Ahead Vhsomc11agmbsue_l0a8cro_yt0858h0s.ami 11gmse_08co_t08582.ai

12. People often describe the height of a mountain as its elevation. How might you
describe an angle of elevation in geometry?

13. You see the prefix tri- in many words, such as triad, triathlon, trilogy, and trimester.
What does the prefix indicate in these words? What geometric figure do you think is
associated with the phrase trigonometric ratio? Explain.

14. You can use the Pythagorean Theorem to derive the Law of Cosines. What do you
think you can find by using the Law of Cosines?

Chapter 8  Right Triangles and Trigonometry 487

CHAPTER Right Triangles

8 and Trigonometry

Download videos VIDEO Chapter Preview 1 Measurement
connecting math Essential Question  How do you find a
to your world.. 8-1 The Pythagorean Theorem and side length or angle measure in a right
Its Converse triangle?
Interactive! ICYNAM
Vary numbers, ACT I V I TI 8-2 Special Right Triangles 2 Similarity
graphs, and figures D 8-3 Trigonometry Essential Question  How do trigonometric
to explore math ES 8-4 Angles of Elevation and Depression ratios relate to similar right triangles?
concepts.. 8-5 Law of Sines
8-6 Law of Cosines DOMAINS
• Similarity, Right Triangles, and Trigonometry
The online • Modeling with Geometry
Solve It will get
you in gear for
each lesson.

Math definitions VOC ABUL ARY Vocabulary
in English and
Spanish English/Spanish Vocabulary Audio Online:

English Spanish

Online access angle of ángulo de depresión
to stepped-out depression, p. 516
problems aligned
to Common Core angle of elevation, p. 516 ángulo de elevación
Get and view
your assignments cosine, p. 507 coseno
online.
Law of Cosines, p. 526 Ley de cosenos

O NLINE Law of Sines, p. 522 Ley de senos
RKME WO
HO Pythagorean triple, p. 492 tripleta de Pitágoras

sine, p. 507 seno

tangent, p. 507 tangente

Extra practice
and review
online

Virtual NerdTM
tutorials with
built-in support

PERFORMANCE TASK

Common Core Performance Task

Locating a Forest Fire

Rangers in the two lookout towers at a state forest notice a plume of smoke, shown
at point C in the diagram. The towers are 2000 m apart. One ranger observes the
smoke at an angle of 54°. The other ranger observes it at an angle of 30°. Both
angles are measured from the line that connects the two towers. When the rangers
call to report the fire, they must state the location of the fire using distances to the
north and west of Lookout Tower B.

CN

b za WE
S

Lookout 54° 30° Lookout
Tower A Tower B
2000 m

Task Description

Find how far to the north and west of Lookout Tower B the fire is located. Round
your answers to the nearest tenth of a meter.

Connecting the Task to the Math Practices MATHEMATICAL

As you complete the task, you’ll apply several Standards for Mathematical PRACTICES
Practice.

• You’ll use trigonometric ratios to relate lengths in the diagram. (MP 1)

• You’ll solve trigonometric equations to find unknown distances. (MP 6)

• You’ll verify results by another method and reason abstractly to write an
equation. (MP 1, MP 2)

Chapter 8  Right Triangles and Trigonometry 489

Concept Byte The Pythagorean MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Theorem
Use With Lesson 8-1 Prepares for GM-ASFRST.9B1.42 .GPr-oSvReTt.h2e.4o rePmrosvaebout
thrieaonrgelmess.a.b.othuet tPryiathnaggleosre. a. n. tThheePoyrethmag. o. r.ean
Activity TMhPeo7rem . . .
MP 7

You will learn the Pythagorean Theorem in Lesson 8-1. The activity below will help you
understand why the theorem is true.

Step 1 Using graph paper, draw any rectangle and label the width a and the b a
length b. ac

Step 2 Cut four rectangles with width a and length b from the graph paper. Then b
cut each rectangle on its diagonal, c, forming eight congruent triangles.

Step 3 Cut three squares from colored paper, one with sides of length a, one with
sides of length b, and one with sides of length c.

hsm11gmse_0801a_t08529.ai

b2 c2
a2

Step 4 Separate the 11 pieces into two groups. ac
b
Group 1: four triangles and the two smaller squares

hGsrmou1p12g: fmousret_ri0a8n0gl1eas_atn0d8t5h3e0la.argiest square

Step 5 Arrange the pieces of each group to form a square.

1. a. How do the areas of the two squares you formed in Step 5 compare?
b. Write an algebraic expression for the area of each of these squares.
c. What can you conclude about the areas of the three squares you cut

from colored paper? Explain.
d. Repeat the activity using a new rectangle and different a and b values.

What do you notice?

2. a. Express your conclusion as an algebraic equation.
b. Use a ruler with any rectangle to find actual measures for a, b, and c.

Do these measures confirm your equation in part (a)?

3. Explain how the diagram at the right represents your equation in Question 2.

4. Does your equation work for nonright triangles? Explore and explain.

hsm11gmse_0801a_t08531.ai

490 Concept Byte  The Pythagorean Theorem

8-1 The Pythagorean Theorem MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

and Its Converse GM-ASFRST.9C1.82 .GU-sSeR. T. .3t.h8e  PUysteha. g. o. trheaenPTyhtheaogreomreaton
Tsohlevoererimghttotrsioalnvgelerisgihnt atrpiapnligedlesprionbalepmplsie. dAlso
Gpr-oSbRleTm.Bs.4Also MAFS.912.G-SRT.2.4

MP 1, MP 3, MP 4, MP 8

Objective To use the Pythagorean Theorem and its converse

Can you use the The squares below fit into groups of three to satisfy the
results of this following equation.
activity to make a area of square 1 + area of square 2 = area of square 3
conjecture about
triangles? Using each square only once, write an equation for each group. What is
the relationship between the three sets of numbers? Explain.
MATHEMATICAL
46 1.5 8 3
PRACTICES 2 2.5
10
5

Lesson The equations in the Solve It demonstrate an important relationship in right triangles
called the Pythagorean Theorem. This theorem is named for Pythagoras, a Greek
Vocabulary mathematician who lived in the 500s b.c. We now know that the Babylonians,
• Pythagorean Egyptians, and Chinese were aware of this relationship before its discovery
by Pythagoras. There are many proofs of the Pythagorean Theorem. You
triple will see one proof in this lesson and others later in the book.

Essential Understanding  If you know the lengths of any two sides of a right
triangle, you can find the length of the third side by using the Pythagorean Theorem.

Theorem 8-1  Pythagorean Theorem

Theorem If . . . Then . . .
If a triangle is a right △ABC is a right triangle (leg1)2 + (leg2)2 = (hypotenuse)2
triangle, then the sum a2 + b 2 = c 2
of the squares of the B
lengths of the legs is ca You will prove Theorem 8-1 in Exercise 49.
equal to the square
of the length of the A bC
hypotenuse.

hsm11gmse_0801_t07530.ai 491

Lesson 8-1  The Pythagorean Theorem and Its Converse

A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the
equation a2 + b2 = c 2. Below are some common Pythagorean triples.

3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

If you multiply each number in a Pythagorean triple by the same whole number, the
three numbers that result also form a Pythagorean triple. For example, the Pythagorean
triples 6, 8, 10, and 9, 12, 15 each result from multiplying the numbers in the triple 3, 4, 5
by a whole number.

Problem 1 Finding the Length of the Hypotenuse

What is the length of the hypotenuse of △ABC? Do the side lengths of  B
△ABC form a Pythagorean triple? Explain. 21
A 20 C
(leg1)2 + (leg2)2 = (hypotenuse)2 Pythagorean Theorem
a2 + b2 = c2 hsm11gmse_0801_t07531.ai

212 + 202 = c2 Substitute 21 for a and 20 for b.

Is the answer 441 + 400 = c2 Simplify.
reasonable?
Yes. The hypotenuse is 841 = c2
the longest side of a right
triangle. The value for c = 29 Take the positive square root.
c, 29, is greater than 20
and 21. The length of the hypotenuse is 29. The side lengths 20, 21, and 29 form a Pythagorean
triple because they are whole numbers that satisfy a2 + b2 = c 2.

Got It? 1. a. The legs of a right triangle have lengths 10 and 24. What is the length of
the hypotenuse?

b. Do the side lengths in part (a) form a Pythagorean triple? Explain.

Problem 2 Finding the Length of a Leg

Which side lengths Algebra  What is the value of x? Express your answer in simplest radical form.
do you have?
a2 + b2 = c2 Pythagorean Theorem 20 8
Remember from
82 + x2 = 202 Substitute.
Chapter 4 that the side
opposite the 90° angle 64 + x2 = 400 Simplify. x
is always the hypotenuse.
x2 = 336 Subtract 64 from each side.
So you have the lengths
x = 1336 Take the positive square root.
of the hypotenuse and x = 116(21) Factor out a perfect square.

one leg.

hsm11gmse_0801_t07532.ai

x = 4 121 Simplify.

Got It? 2. The hypotenuse of a right triangle has length 12. One leg has length 6. What
is the length of the other leg? Express your answer in simplest radical form.

492 Chapter 8  Right Triangles and Trigonometry

Problem 3 Finding Distance

Dog Agility  Dog agility courses often contain a seesaw obstacle, as shown below. To
the nearest inch, how far above the ground are the dog’s paws when the seesaw is
parallel to the ground?

36 in.

26 in.

a2 + b2 = c2 Pythagorean Theorem

How do you know 262 + b2 = 362 Substitute.
when to use a
calculator? 676 + b2 = 1296 Simplify.
This is a real-world
situation. Real-world b2 = 620 Subtract 676HfSroMm11eGacMhSsEid_e0.801_a08325
distances are not usually b ≈ 24.8997992 Use a calculapthoor ttoo pta0k8e3t2h4e positive square root.
expressed in radical form.
The dog’s paws are 25 in. above the ground.

Got It? 3. The size of a computer monitor is the length of its diagonal. You want to buy
a 19-in. monitor that has a height of 11 in. What is the width of the monitor?
Round to the nearest tenth of an inch.

You can use the Converse of the Pythagorean Theorem to determine whether a triangle
is a right triangle.

Theorem 8-2  Converse of the Pythagorean Theorem

Theorem If . . . a2 + b2 = c2 Then . . .
If the sum of the squares of B △ABC is a right triangle
the lengths of two sides of a
triangle is equal to the square ca
of the length of the third side,
then the triangle is a right A bC
triangle.
You will prove Theorem 8-2 in Exercise 52.

hsm11gmse_0801_t07530.ai 493

Lesson 8-1  The Pythagorean Theorem and Its Converse

Problem 4 Identifying a Right Triangle

How do you know A triangle has side lengths 85, 84, and 13. Is the triangle a right triangle? Explain.
where each of the
side lengths goes in a2 + b2 ≟ c2 Pythagorean Theorem
the equation?
132 + 842 ≟ 852 Substitute 13 for a, 84 for b, and 85 for c.
Work backward. If
169 + 7056 ≟ 7225 Simplify.
the triangle is a right
7225 = 7225 ✓
triangle, then the

hypotenuse is the longest Yes, the triangle is a right triangle because 132 + 842 = 852.
side. So use the greatest

number for c.

Got It? 4. a. A triangle has side lengths 16, 48, and 50. Is the triangle a

right triangle? Explain.

b. Reasoning  Once you know which length represents the hypotenuse,

does it matter which length you substitute for a and which length you

substitute for b? Explain.

The theorems below allow you to determine whether a triangle is acute or obtuse. These
theorems relate to the Hinge Theorem, which states that the longer side is opposite the
larger angle and the shorter side is opposite the smaller angle.

Theorem 8-3 If . . . Then . . .
c2 7 a2 + b2 △ABC is obtuse
Theorem
If the square of the length of the B
longest side of a triangle is greater
than the sum of the squares of the ac
lengths of the other two sides, then
the triangle is obtuse. C bA

Theorem 8-4 You will prove Theorem 8-3 in Exercise 53.

Theorem Ihf s. m. .11gmse_0801_t0753T3h.aein . . .
If the square of the length of the c2 6 a2 + b2
longest side of a triangle is less △ABC is acute
than the sum of the squares of the
lengths of the other two sides, then B
the triangle is acute.
ac

C bA

You will prove Theorem 8-4 in Exercise 54.

hsm11gmse_0801_t07534.ai

494 Chapter 8  Right Triangles and Trigonometry

What information do Problem 5 Classifying a Triangle
you need?
You need to know how A triangle has side lengths 6, 11, and 14. Is it acute, obtuse, or right?
the square of the longest
side compares to the c2 ■ a2 + b2 Compare c2 to a2 + b2.
sum of the squares of the 142 ■ 62 + 112 Substitute the greatest value for c.
other two sides. 196 ■ 36 + 121 Simplify.
196 7 157
Since c 2 7 a2 + b 2, the triangle is obtuse.

Got It? 5. Is a triangle with side lengths 7, 8, and 9 acute, obtuse, or right?

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES

What is the value of x in simplest radical form? 5. Vocabulary  Describe the conditions that a set

1. x 12 2. of three numbers must meet in order to form
7x a Pythagorean triple.
35
6. Error Analysis  A triangle
9 162 + 342 =? 302
has side lengths 16, 34, and

3. x 4. 13 30. Your friend says it is not 256 + 1156 =? 900
a right triangle. Look at your 1412 ≠ 900
x
3hsm115gmse_0801_t0753h5s.ami 1111gmse_0801_t07536.ai friend’s work and describe
the error.

hsm11gmse_0801_t07539.ai

hsPmr1a1cgtmicsee_a08n0d1_Pt0r7o5b3h7les.ammi 1-1Sgomlsvei_n0g801E_xt0e7r5c3i8s.eais MATHEMATICAL

PRACTICES

A Practice Algebra  Find the value of x. See Problem 1.

7. 8 8. x 7 9. x
30
6x 24 16

10. 11. 12. 15
x 12
hsmx 11gm72se_0801_t07541.ai 8x
hsm1161gmse_0801_t07540.ai
65 hsm11gmse_0801_t07542.ai

Does each set of numbers form a Pythagorean triple? Explain.

13. 4, 5, 6 14. 10, 24, 26 15. 15, 20, 25
hsm11gmse_0801_t07545.ai
hsm11gmse_0801_t07543.ai hsm11gmse_0801_t07544.ai

Lesson 8-1  The Pythagorean Theorem and Its Converse 495

Algebra  Find the value of x. Express your answer in simplest radical form. See Problem 2.

16. 17. x 18. 15 x
7 18
64 4

x

19. 20. 21.

19 x x6 hs1m0 11gmx se_0801_t07548.ai

hsm11gmse_0801_t07546.ai hsm11gmse_0801_t07547.ai

16 x 5

22. Home Maintenance  A painter leans a 15-ft ladder against a house. See Problem 3.

Tfhohosemt,bh1ao1swegmohfsitgehh_e0ol8and0td1he_erth0ios7u55s4fet9fd.raooimes the house. To the nearest tenth of a hsm11gmse_0801_t07551.ai
the lhadsmde1r1rgemacshe?_0801_t07550.ai

23. A walkway forms one diagonal of a square playground. The walkway is 24 m long.

To the nearest meter, how long is a side of the playground?

Is each triangle a right triangle? Explain. See Problem 4.

24. 25. 8 25 26. 65
24 56
20 28 33

19

The lengths of the sides of a triangle are given. Classify each triangle as acute, See Problem 5.

right, or obtuse. hsm11gmse_0801_t07553.ai

27. 4h,s5m, 61 1gmse_0801_t07552.ai 28. 0.3, 0.4, 0.6 29. 1h1s,m121,11g5mse_0801_t07554.ai

30. 13, 2, 3 31. 30, 40, 50 32. 111, 17, 4

B Apply 33. Think About a Plan  You want to embroider a square x
design. You have an embroidery hoop with a 6-in. x
diameter. Find the largest value of x so that the entire
square will fit in the hoop. Round to the nearest tenth.
• What does the diameter of the circle represent in
the square?
• What do you know about the sides of a square?
• How do the side lengths of the square relate to the
length of the diameter?

34. In parallelogram RSTW, RS = 7, ST = 24, and RT = 25.
Is RSTW a rectangle? Explain.

496 Chapter 8  Right Triangles and Trigonometry

35. Coordinate Geometry  You can use the Pythagorean Theorem to prove the

Proof Distance Formula. Let points P(x1, y1) and Q(x2, y2) be the endpoints of the
hypotenuse of a right triangle.

a. Write an algebraic expression to complete each of the y Q(x2, y2)
O
following: PR = ? and QR = ? . R(x2, y1)
b. By the Pythagorean Theorem, PQ2 = PR2 + QR2. Rewrite x

this statement by substituting the algebraic expressions you

found for PR and QR in part (a).

c. Complete the proof by taking the square root of each side of P(x1, y1)

the equation that you wrote in part (b).

Algebra Find the value of x. If your answer is not an integer, express it in
simplest radical form.

36. 26 x 26 37. x 38.h sm11gmse3_0801_t07558.ai
16
48 4 V5 2x
4
3

For each pair of numbers, find a third whole number such that hsm4121.g 1m2s, e3_70801_t07557.ai
the thhrseme 1n1ugmmbseer_s0fo8r0m1_at0P7y5th55ag.aoirean triple.

39. 20, 21 40. 14, 48 hsm11gmse_4018. 0113_,t8057 556.ai

Open-Ended  Find integers j and k such that (a) the two given integers and j
represent the side lengths of an acute triangle and (b) the two given integers
and k represent the side lengths of an obtuse triangle.

43. 4, 5 44. 2, 4 45. 6, 9 46. 5, 10 47. 6, 7 48. 9, 12

49. Prove the Pythagorean Theorem. C a B
Proo f Given:  △ABC is a right triangle. r
b c
Prove:  a2 + b2 = c 2 q
(Hint: Begin with proportions suggested by Theorem 7-3 or
AD
its corollaries.)

STEM 50. Astronomy  The Hubble Space Telescope orbits 600 km above Earth’s 600 km x
surface. Earth’s radius is about 6370 km. Use the Pythagorean Theorem
to find the distance x from the telescope to Earth’s horizon. Round your hsm11gmse_08016_37t00 7k5m59.ai
answer to the nearest ten kilometers. (Diagram is not to scale.)

51. Prove that if the slopes of two lines have product -1, then the lines are
perpendicular. Use parts (a)–(c) to write a coordinate proof.

a. First, argue that neither line can be horizontal or vertical.

b. Then, tell why the lines must intersect. (Hint: Use indirect reasoning.)

c. Place the lines in the coordinate plane. Choose a point on /1 and find a related
point on /2. Complete the proof.
hsm11gmse_0801_t07560.ai

Lesson 8-1  The Pythagorean Theorem and Its Converse 497

C Challenge 52. Use the plan and write a proof of Theorem 8-2 (Converse of the B
Proof Pythagorean Theorem). ca
A bC
Given:  △ABC with sides of length a, b, and c, where a2 + b2 = c 2
Prove:  △ABC is a right triangle. hsm11gmse_0801_t07561.ai
B
Plan:  Draw a right triangle (not △ABC) with legs of lengths a and b. Label
the hypotenuse x. By the Pythagorean Theorem, a2 + b2 = x 2. Use ac
substitution to compare the lengths of the sides of your triangle and C bA
△ABC. Then prove the triangles congruent.
B
53. Use the plan and write a proof of Theorem 8-3. hsma11gmse_0c801_t08128.ai
C bA
Proof of length a, b, and c, where c2 7 a2 + b2

Given:  △ABC with sides

Prove:  △ABC is an obtuse triangle.

Plan:  D raw a right triangle (not △ABC) with legs of lengths a and b.
Label the hypotenuse x. By the Pythagorean Theorem, a2 + b2 = x 2.

Use substitution to compare lengths c and x. Then use the Converse

of the Hinge Theorem to compare ∠C to the right angle.

54. Prove Theorem 8-4.
Proo f Given:  △ABC with sides of length a, b, and c, where c 2 6 a2 + b2

Prove:  △ABC is an acute triangle.

Standardized Test Prep hsm11gmse_0801_t08129.ai

SAT/ACT 55. A 16-ft ladder leans against a building, as shown. To the nearest foot, Ladder
how far is the base of the ladder from the building?

56. What is the measure of the complement of a 67° angle? 15.5 ft

57. The measure of the vertex angle of an isosceles triangle is 58.
What is the measure of one of the base angles?

58. The length of rectangle ABCD is 4 in. The length of similar rectangle DEFG is
6 in. How many times greater than the area of ABCD is the area of DEFG?

hsm11gmse_0801_t11734

Mixed Review See Lesson 7-5.

59. △ABC has side lengths AB = 8, BC = 9, and AC = 10. Find the lengths of
the segments formed on BC by the bisector of ∠A.

Get Ready!  To prepare for Lesson 8-2, do Exercises 60–62.

Simplify each expression. See Review, p. 399.

60. 19 , 13 61. 30 , 12 62. 16
13

498 Chapter 8  Right Triangles and Trigonometry

8-2 Special Right Triangles CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

MG-ASFRST.9C1.82 .GU-sSeR. T. .3t.h8e  PUysteha. g. o. trheaenPTyhtheaogreomreaton
Tsohlevoererimghttotrsioalnvgelerisgihnt atrpiapnligedlesprionbalepmplsie. d
pMroPb1le,mMs.P 3, MP 4
MP 1, MP 3, MP 4

Objective To use the properties of 45°-45°-90° and 30°-60°-90° triangles

There are a lot This map of part of a college campus shows Student Dorm
of similar right a square “quad” area with walking paths. The Center Dining Hall
triangles here. In distance from the dorm to the dining hall is Library
the lesson, you’ll 150 yd. Science
learn a shortcut Lab
for finding some Suppose you go from your dorm to the dining
of these distances. hall, to the science lab, to your dorm, to the
student center, to the library, and finally
MATHEMATICAL back to your dorm. To the nearest tenth,
how far do you walk? Justify your answer.
PRACTICES (Assume you always take the most direct
routes and stay on the paths.)

The Solve It involves triangles with angles 45°, 45°, and 90°.

Essential Understanding  Certain right triangles have properties that allow
you to use shortcuts to determine side lengths without using the Pythagorean Theorem.

The acute angles of a right isosceles triangle are both 45° angles. Another name for
an isosceles right triangle is a 45°-45°-90° triangle. If each leg has length x and the

hypotenuse has length y, you can solve for y in terms of x.

x2 + x2 = y2 Use the Pythagorean Theorem. xx
2x2 = y2 Simplify. 45Њ 45Њ
x 12 = y Take the positive square root of each side.
y

You have just proved the following theorem.

Theorem 8-5  45°-45°-90° hsm11gmse_0802_t07572.ai
Triangle Theorem

In a 45°-45°-90° triangle, both legs are congruent and the s V2 45Њ s
length of the hypotenuse is 12 times the length of a leg.
45Њ
#hypotenuse = 12 leg s

Lesson 8-2  Special Right Triangles 499

Problem 1 Finding the Length of the Hypotenuse

Why is only one leg What is the value of each variable?
labeled?
A 45°-45°-90° triangle is A B 45Њ
a right isosceles triangle, 9 x 45Њ 2 V2
so the legs have equal
lengths. 45Њ 45Њ
h

# #hypotenuse = 12 leg 45°-45°-90° △ Theorem hypotenuse = 12 leg
# #hsm11gm she=_018022_9t0 7574.ai Substitute.
 h = 912 Simplify. x = 12 2 12
hsm11gmsex_=08402_t07575.ai

Got It? 1. What is the length of the hypotenuse of a 45°-45°-90° triangle
with leg length 513?

Problem 2 Finding the Length of a Leg

Can you eliminate Multiple Choice  What is the value of x?

any of the choices? 3 6 x 45Њ 6

tT hhee variable x represents 3 12 6 12 45Њ
length of a leg. Since 45°-45°-90° Triangle Theorem
#hypotenuse = 12 leg Substitute.
the hypotenuse is the #6 = 12 x

longest side of a right

triangle, x 6 6. You
can eliminate choices
#x = 162 Divide each side by 12.
C and D. hsm11gmse_0802_t07576.ai

x = 6 12 Multiply by a form of 1 to rationalize the denominator.
12 12

x = 6 12 2 Simplify.

x = 312 Simplify.

The correct answer is B.

Got It? 2. a. The length of the hypotenuse of a 45°-45°-90° triangle is 10. What is the

length of one leg? 6 1122 ?
12
b. Reasoning  In Problem 2, why can you multiply by

500 Chapter 8  Right Triangles and Trigonometry

When you apply the 45°-45°-90° Triangle Theorem to a real-life example, you can use a
calculator to evaluate square roots.

Problem 3 Finding Distance

How do you Softball  A high school softball diamond is a square. The distance 60 ft
know that d is a from base to base is 60 ft. To the nearest foot, how far does a catcher d
hypotenuse? throw the ball from home plate to second base?
The diagonal d is part
of two right triangles. The distance d is the length of the hypotenuse of a 45°-45°-90° triangle.
The hypotenuse of a #hypotenuse = 12 leg
right triangle is always d = 6012
opposite the 90° angle.
So d must be a d ≈ 84.85281374 Use a calculator.
hypotenuse.
The catcher throws the ball about 85 ft from home plate to second base. hsm11gmse_0802_t07577.ai

Got It? 3. You plan to build a path along one diagonal of a 100 ft-by-100 ft square
garden. To the nearest foot, how long will the path be?

Another type of special right triangle is a 30°-60°-90° triangle.

Theorem 8-6  30°-60°-90° Triangle Theorem 2s 30Њ s V3
60Њ
In a 30°-60°-90° triangle, the length of the hypotenuse is
twice the length of the shorter leg. The length of the longer s
leg is 13 times the length of the shorter leg.

##hypotenuse = 2 shorter leg

longer leg = 13 shorter leg

Proof Proof of Theorem 8-6: 30°-60°-90° Triangle Theorem hsm11gmWse_0802_t07578

For equilateral △WXZ, altitude WY bisects ∠W and is the 30Њ

perpendicular bisector of XZ. So, WY divides △WXZ into
two congruent 30°-60°-90° triangles.

Thus, XY = 1 XZ = 1 XW , or XW = 2XY = 2s.
2 2

XY 2 + YW 2 = XW 2 Use the Pythagorean Theorem. 60Њ 60Њ Z
X sY
s2 + YW 2 = (2s)2 Substitute s for XY and 2s for XW.

YW 2 = 4s2 - s2 Subtract s2 from each side.

YW 2 = 3s2 Simplify. hsm11gmse_0802_t07579

YW = s13 Take the positive square root of each side.

Lesson 8-2  Special Right Triangles 501

You can also use the 30°-60°-90° Triangle Theorem to find side lengths.

Problem 4 Using the Length of One Side d5
Algebra  What is the value of d in simplest radical form? 60Њ 30Њ
f

In a 30°-60°-90° triangle, the #longer leg = 13 shorter leg hsm11gmse_0802_t07580
leg opposite the 60° angle is the
longer leg. So d represents the 5 = d 13

length of the shorter leg. Write

an equation relating the legs.

Divide each side by 13 to solve d = 5
for d. 13

The value of d is not in simplest #5 13 = 513
radical form because there is a 13 3
radical in the denominator. Multiply 13
d by a form of 1.
So d = 5 13 3.

Got It? 4. In Problem 4, what is the value of f   in simplest radical form?

Problem 5 Applying the 30°-60°-90° Triangle Theorem

How does knowing Jewelry Making  An artisan makes pendants in the shape of
the shape of the equilateral triangles. The height of each pendant is 18 mm.
pendants help? What is the length s of each side of a pendant to the nearest
Since the triangle is tenth of a millimeter?
equilateral, you know
that an altitude divides The hypotenuse of each 30°-60°-90° triangle is s. The shorter
the triangle into two
congruent 30°-60°-90° leg is 1 s. #longer leg = 13 shorter leg
triangles. 2

( )18 = 131 s s
2 18 mm

# 18 = 13 s Simplify.
2 Multiply each side by 123.
Use a calculator.
2 18 = s
13
s ≈ 20.78460969

Each side of a pendant is about 20.8 mm long.

Got It? 5. Suppose the sides of a pendant are 18 mm long. What is the height of the
pendant to the nearest tenth of a millimeter?

502 Chapter 8  Right Triangles and Trigonometry

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES

What is the value of x? If your answer is not an integer, 5. Error Analysis  Sandra drew the triangle below. Rika
express it in simplest radical form. said that the labeled lengths are not possible. With
which student do you agree? Explain.
1. 2.
5 5 ͙3
x 45Њ 7 30Њ x 30° 60°
60Њ
6
45Њ

3. 45Њ 4. 10

hx s4m5Њ11g8 mse_0802_t0758h1sm11gxmse12_0802_t07582 6. Reasoning  A test question asks you to find two side
lengths of a 45°-45°-90° triangle. You know that
60Њ 60Њ
ftohremleunlagtfhorof4ho5s°nm-e415le°1g-g9is0m°6st,rebia_unt0gy8loe0us2.f_Eotxr0gpo7lat5itn9he3hospweycoiaul

can still determine the other side lengths. What are

hsm11gmse_0802_t07583hsm11gmse_0802_t11761 the other side lengths?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find the value of each variable. If your answer is not an integer, express it See Problems 1 and 2.
in simplest radical form.
y 60
7. 8. y 9. 45Њ
8 45Њ y V2 45Њ
x
45Њ
x

10. 11. 12. V5 V5

hy sm11g15mVs2e_0802_t07586 hs45mЊ 111g0mse_0802_t07588 hsm1y1gmse_0802_t07589

45Њ 45Њ
x
x

13. ywDhoihsnuimncrhea1r na1ws2gwa0mre-ercsmtDeoe_ctsh0hiego8npn0 se2taWi_crtekh0sact7tat5ines9nftit0htheaolsofidnahegcselaemndntigi1amt1gheogotnmefart.lshwee_ist0mh8oau0llt2eas_nttys0qo7uv5ae9rreh1apnlagt?ehRosonmun1d1gmse_S0e8e0P2r_otb0l7e5m932.

14. Aviation  The four blades of a helicopter meet at right angles and are all the same
length. The distance between the tips of two adjacent blades is 36 ft. How long is
each blade? Round your answer to the nearest tenth of a foot.

Lesson 8-2  Special Right Triangles 503

Algebra  Find the value of each variable. If your answer is not an integer, See Problems 4 and 5.
express it in simplest radical form.
30Њ
15. 16. y 17. 10
40 x x y
60Њ 30Њ 60Њ
30Њ 2 V3 x
y

18. 19. hysm11gm2seV_30802_t07595 20. 9 V3
x 60Њ 12
60Њ 30Њ x
hsm11gmse_0802_t07594 x
hsm11gymse_600Њ802_t07596
y

STEM 21. Architecture  An escalator lifts people to the second floor of a building,
25 ft above the first floor. The escalator rises at a 30° angle. To the

nhesamre1st1fgoomt,sheo_w08fa0r2d_ote0s7a5p9e7rson trhasvmel 1fr1ogmmthseeb_o0t8to0m2_tot0th7e59to8p hsm11gmse_0802_t072559f9t
of the escalator?
30Њ

STEM 22. City Planning  Jefferson Park sits on one square city block 300 ft on

each side. Sidewalks across the park join opposite corners. To the nearest foot,

how long is each diagonal sidewalk?

B Apply Algebra Find the value of each variable. If your answer is not an integer, hsm11gmse_0802_t07600
express it in simplest radical form.

23. 7 V2 a b 24. 25. a 10
45Њ 30Њ 30Њ 60Њ
cd 4 V3 a b b

60Њ 45Њ cd
c d

26. 6 27. 4 28. 8

hsm11gmse_6008Њ 02_t07601 3 V2 a 6hsm11gmse_0b802_t07603

2 V3 a hsm4151Њ gmse_0802_t07602 45Њ
a
b b

29. Think About a Plan  A farmer’s conveyor belt carries

bchoasnlmevse1oy1fohgr abmyelsftreom_mo0vt8he0se2ag_trot10u070n6dft0/tom4tihne. Hbahorwsnmmlo1aft1n.gyThmese_0802_t07605 hsm11gmse_0802_t07606

seconds does it take for a bale of hay to go from the

ground to the barn loft?

• Which part of a right triangle does the conveyor 12 ft
belt represent?

• You know the speed. What other information do

you need to find time? 30؇
• How are minutes and seconds related?

504 Chapter 8  Right Triangles and Trigonometry HSM11GMSE_0802_a08330
2nd pass 01-12-09
Durke

30. House Repair  After heavy winds damaged a house, workers placed a 6-m brace 30Њ
against its side at a 45° angle. Then, at the same spot on the ground, they placed a 6m
second, longer brace to make a 30° angle with the side of the house.
45Њ
a. How long is the longer brace? Round to the nearest tenth of a meter.
hsmd11gms2e_0802_
b. About how much higher does the longer brace reach than the shorter brace?
2
31. Open-Ended  Write a real-life problem that you can solve using a 30°-60°-90° 2
triangle with a 12-ft hypotenuse. Show your solution.

32. Constructions  Construct a 30°-60°-90° triangle using a segment that is

the given side.

a. the shorter leg b. the hypotenuse c. the longer leg

C Challenge 33. Geometry in 3 Dimensions  Find the length d, in simplest radical d1
form, of the diagonal of a cube with edges of the given length. 11

a. 1 unit b. 2 units c. s units

Standardized Test Prep

SAT/ACT 34. The longer leg of a 30°-60°-90° triangle is 6. What is the length of the hypotehnsumse1?1gmse_0802_t07608

2 13 3 12 4 13 12

35. Which triangle is NOT a right triangle?



45 5 V2 0.6 0.8 4 V5 8 V5
36
5

27 1.0 7 V5
5

36. Suppose p is false and q is true. Which statement is NOT true?

37. In rig phhtsS△mAq1B 1Cg,m∠sCei_s0th8e0r2i_ght0t ∼ha7qns6mg0¡le91a(1pngdm¿CsDqe)_i s0t8h0e2a_ltitt0 uph7ds6e¡m1d0q1ra 1wgnmtoseth_e0802_t0 (h7ps6m¡121q1)g¿m∼sep_0802_t07611
Short
Response hypotenuse. If AD = 3 and DB = 9, what is AC? Show your work.

Mixed Review See Lesson 8-1.

38. A right triangle has a 6-in. hypotenuse and a 5-in. leg. Find the length of the
other leg in simplest radical form.

39. An isosceles triangle has 20-cm legs and a 16-cm base. Find the length of the
altitude to the base in simplest radical form.

Get Ready!  To prepare for Lesson 8-3, do Exercises 40–43.

Algebra  Solve each proportion. See Lesson 7-1.

40. 3x = 74 41. 6 = x 42. 8 = x4 43. 5x = 7
11 9 15 12

Lesson 8-2  Special Right Triangles 505

Concept Byte Exploring MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Trigonometric Ratios
Use With Lesson 8-3 MG-ASFRST.9C1.62 .GU-nSdRerTs.t3a.n6d  Uthnadtebrsytasinmdiltahraitty,bsyide
sraimtioilasriintyr,isgihdtetraiatniogsleins arirgehpt rtoripaenrgtiles oafrethe
technology panrogplesrtinesthoef thrieanagnlgel,elseainditnhgettoriadnegfilnei,tiloeandsionfg
torigdoenfionmitieotnrisc oraf ttiroigsofonromaceuttreic arnagtiloes.for
aMcuPte5 angles.
MP 5

Construct
Ua speoginetoDmoentryAsBo>,ftcwoanrsetrtuocctoanlsintreupcterApBe>nadnidcuAlaCr>
so AthBa>tth∠aAt iinstaecrsuetec.tsThArCo>uingh C
to E
DB
point E.

Moving point D changes the size of △ADE. Moving point C changes the size A
of ∠A.

Exercises

1. • Measure ∠A and find the lengths of the sides of △ADE. hsm11gmse_0803a_t08533.ai

• Calculate the ratio leg opposite ∠A , which is ED . C
hypotenuse AE E
• Move point D to change the size of △ADE without changing m∠A. Leg
Hypotenuse opposite ЄA

What do you observe about the ratio as the size of △ADE changes?

2. • Move point C to change m∠A. A Leg adjacent D B
a. What do you observe about the ratio as m∠A changes? to ЄA
b. What value does the ratio approach as m∠A approaches 0? As m∠A

approaches 90?

3. • Make a table that shows values for m∠A and the ratio leg opposite ∠A .
In your table, include 10, 20, 30, c , 80 for m∠A. hypotenuse

• Compare your table with a table of trigonometric ratios. hsm11gmse_0803a_t08536.ai

Do your values for leg opposite ∠A match the values in one of the columns
hypotenuse

of the table? What is the name of this ratio in the table?

Extend

4. Repeat Exercises 1–3 for leg hadypjaocteenntutsoe∠A , which is AD , and leg opposite∠A ,
AE leg adjacent to∠A
which is AEDD.

5. • Choose a measure for ∠A and determine the ratio r = leghoyppoptoesnitues∠e A.
Record m∠A and this ratio.

• Manipulate the triangle so that leg adjacent to ∠A has the same value r.
hypotenuse
Record this m∠A and compare it with your first value of m∠A.

• Repeat this procedure several times. Look for a pattern in the two
measures of ∠A that you found for different values of r.

Make a conjecture.

506 Concept Byte  Exploring Trigonometric Ratios

8-3 Trigonometry MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

MG-ASFRST.9C1.82 .GU-sSeRtrTi.g3o.n8o  mUesetritcrigraotnioosmaentdricthreatios
aPnytdhtahgeorPeyatnhaTghoeroeraenmTthoeosorelvme troigshotltvreiarnigghlets
itnri aanpgplleiesdinp raopbplleiemds.pArolbsloe,mGs-.SARlsTo.C,.7, G-MG.A.1
MMPAF1S,.M91P2.3G, -MSRPT4.3, .M7,PM6AFS.912.G-MG.1.1
MP 1, MP 3, MP 4, MP 6

Objective To use the sine, cosine, and tangent ratios to determine side lengths and
angle measures in right triangles

What is the ratio of the length of the shorter leg to the length of the

hypotenuse for each of △ADF, △AEG, and △ABC? Make a conjecture

based on your results. B

Here are ratios E
in triangles once
again! This must D 4
be “similar” to A 2F 6 G4C
something you’ve
seen before.

MATHEMATICAL

PRACTICES

Lesson Essential Understanding  If you know certain combinations of side lengths
and angle measures of a right triangle, you can use ratios to find other side lengths and
Vocabulary angle measures.
• trigonometric
Any two right triangles that have a pair of congruent acute angles are similar by
ratios the AA Similarity Postulate. Similar right triangles have equivalent ratios for their
• sine corresponding sides called trigonometric ratios.
• cosine
• tangent Key Concept  Trigonometric Ratios

sine of  ∠A = length of leg opposite ∠A = a B
length of hypotenuse c ca
A bC
cosine of  ∠A = length of leg adjacent to ∠A = b
length of hypotenuse c

tangent of  ∠A = length of leg opposite ∠A = a
length of leg adjacent to ∠A b

Lesson 8-3  Trigonometry hsm11gmse_0803_t07622.ai

507

You can abbreviate the ratios as

sin A = hyoppoptoesniutese, cos A = hyapdojatecnenutse , and tan A = opposite .
adjacent

Problem 1 Writing Trigonometric Ratios

How do the sides What are the sine, cosine, and tangent ratios for jT ?
relate to jT ?
GR is across from, or sin T = opposite = 8 G
opposite, ∠T . TR is next hypotenuse 17 8
to, or adjacent to, ∠T . 17 R
TG is the hypotenuse cos T = adjacent = 15 T 15
because it is opposite the hypotenuse 17
90° angle.
tan T = opposite = 8
adjacent 15
What is the first
step? Got It? 1. Use the triangle in Problem 1. What are the sine, cosine, and tangent ratios
Look at the triangle and for ∠G?
determine how the sides hsm11gmse_0803_t07623.ai
of the triangle relate to
the given angle. In Chapter 7, you used similar triangles to measure distances indirectly.
You can also use trigonometry for indirect measurement.

Problem 2 Using a Trigonometric Ratio to Find Distance

Landmarks  In 1990, the Leaning Tower of
Pisa was closed to the public due to safety
concerns. The tower reopened in 2001
after a 10-year project to reduce its tilt from
vertical. Engineers’ efforts were successful
and resulted in a tilt of 5°, reduced from 5.5°.
Suppose someone drops an object from the
tower at a height of 150 ft. How far from the base
of the tower will the object land? Round to the
nearest foot.

The given side is adjacent to the given angle. The
side you want to find is opposite the given angle.

tan 5° = 15x0 Use the tangent ratio.

x = 150(tan 5°) Multiply each side by 150.

150 tan 5 enter Use a calculator. 5º
150 ft
x ≈ 13.12329953

The object will land about 13 ft
from the base of the tower.

508 Chapter 8  Right Triangles and Trigonometry

Got It? 2. For parts (a)–(c), find the value of w to the nearest tenth.

a. 54Њ b. 1.0 c. w

17 w w 33Њ
28Њ 4.5

d. A section of Filbert Street in San Francisco rises at an angle of about 17°.

Infheyasomrues1wt1afoglkomt1.5s0ef_t0u8p0th3i_sts0e7cht6iso2mn4,.1aw1ihgamt isseyo_u0r8v0e3rt_icta0l7r6is2eh?5sR.amoiu1n1dgtmo tshee_0803_t07626.ai

If you know the sine, cosine, or tangent ratio for an angle, you can use an inverse
 (sin-1, cos-1, or tan-1) to find the measure of the angle.

Problem 3 Using Inverses BX
What is mjX to the nearest degree?
A H

6 10 15 20
BX MN

You know the lengths of the You know the lengths of the

ohhpysppmoots1ei1tnegu∠smeXsa.en_d0th8e0s3i_dte07627.ai hypotenuse and the side

Use the sine ratio. ahdsjmac1en1tgtmo ∠seX_.0803_t07628.ai

Use the cosine ratio.

When should you use sin X = 6 Write the ratio. cos X = 15
an inverse? 10 Use the inverse. 20
Use an inverse when you ( )m∠X = sin-1 ( )m∠X = cos-1
know two side lengths of 6 15
a right triangle and you 10 20
want to find the measure
of one of the acute sin–1 6 10 enter Use a calculator. cos–1 15 20 enter
angles.
m∠X ≈ 36.86989765 m∠X ≈ 41.40962211

≈ 37   ≈ 41

Got It? 3. a. Use the figure at the right. What is m∠Y to P 100 T
41
the nearest degree?
Y
b. Reasoning  Suppose you know the lengths

of all three sides of a right triangle. Does it

matter which trigonometric ratio you use to

find the measure of any of the three angles? Explain.

hsm11gmse_0803_t07629.ai

Lesson 8-3  Trigonometry 509

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES

Write each ratio. 9. Vocabulary  Some people use SOH-CAH-TOA to

1. sin A 2. cos A A remember the trigonometric ratios for sine, cosine,
3. tan A and tangent. Why do you think that word might help?
4. sin B 6 10 (Hint: Think of the first letters of the ratios.)

5. cos B 6. tan B C8B 10. Error Analysis  A student states that sin A 7 sin X

because the lengths of the sides of △ABC are greater

What is the value of x? Round to the nearest tenth. than the lengths of the sides of △XYZ. What is the

7. 8. 27 student’s error? Explain.

x xЊ 32 hsm11gmse_0803_t07630.ai Y B
39Њ
15

35Њ XC 35Њ
Z A

hsm11gmse_0803_t0763h1.sami 11gmse_0803_t07632.ai

Practice and Problem-Solving Exercises PMhRAsTAmHCEM1TAI1TCIgCEAmSL se_0803_t07633.ai

A Practice Write the ratios for sin M, cos M, and tan M. See Problem 1.

11. 25 M 12. M 7 K 13. K 2 V3 L
L 9 4 V2 2
7 24 L M 4
K

Find the value of x. Round to the nearest tenth. See Problem 2.

14. hsm2011gmsxe_0803_t076341.a5i. 16. h7s6m4Њ 11gmxse_0803_t07636.ai

35Њ hsmx41Њ11gmse_0803_t07635.ai

11

17. 18. 28Њ 19. 10
x 10 x 62Њ 50 x

hsm316Њ1gmse_0803_t07637.ai hsm11gmse_0803_t07638.ai hsm1215gЊ mse_0803_t07639.ai

20. Recreation  A skateboarding ramp is 12 in. high and rises at an angle of 17°. How

long is the base of the ramp? Round tho sthme1n1egarmesstein_c0h8. 03_t07641.ai rihsesmof11gmse_0803_t07642.ai

21. Phusbmlic1T1ragnmspsoer_t0at8i0on3 _At0n7e6sc4a0la.atoi r in the subway station has a vertical

195 ft 9.5 in., and rises at an angle of 10.4°. How long is the escalator? Round to

the nearest foot.

510 Chapter 8  Right Triangles and Trigonometry

Find the value of x. Round to the nearest degree. See Problem 3.

22. 5 14 23. 8 24. 13
xЊ xЊ xЊ
5
9

25. 26. xЊ 27. 0.34
3h.0smxЊ11g5m.8se_0803_t07643.ai 17 xЊ 0.15

hsm114g1mse_0803_t07644.ai hsm11gmse_0803_t07645.ai

B Apply 28. The lengths of the diagonals of a rhombus are 2 in. and 5 in. Find the measures of
the angles of the rhombus to the nearest degree.
hsm11gmse_0803_t07648.ai
29. Thhisnmk 1A1bgoumtsaeP_l0a8n0  C3a_rtl0os7p6l4a6n.satio bhusimld 1a1ggraminsbei_n0w8i0th3a_t07647.ai

radius of 15 ft. The recommended slant of the roof is 25°. He x

wants the roof to overhang the edge of the bin by 1 ft. What 25Њ

should the length x be? Give your answer in feet and inches.

• What is the position of the side of length x in relation to 1 ft
over-
the given angle? hang
• What information do you need to find a side length of a

right triangle? 15 ft
• Which trigonometric ratio could you use?

An identity is an equation that is true for all the allowed values of the variable.

Use what you know about trigonometric ratios to show that each equation is

an identity. #31. sin X = cos X tan X hsm11gmse_0803_t07649
30. tan X = csoins XX
32. cos X = sin X
tan X

Find the values of w and then x. Round lengths to the nearest tenth and angle
measures to the nearest degree.

33. 34. 35.
6 10 102
30Њ w xЊ 4 56Њ 34Њ w 102
wx 42Њ x

STEM 36. Pyramids  All but two of the pyramids built by the

ahncsimen1t1Eggmypstiea_n0s 8ha0v3e_fta0c7es6i5n0clined at 52° angles. hsm11gmse_0803_t07652
pSuyrpapmoisde.aMnoasrtcohfatehoelopgyirsatmdiisdcohvaesresrtohhdesemrdu,1ibn1usgtotmfhaese_0803_t07651

archaeologist is able to determine that the length of

a side of the square base is 82 m. How tall was the 52Њ

pyramid, assuming its faces were inclined at 52°?

Round your answer to the nearest meter. 82 m

HSM11GMSE_0803_a08334
2nd pass 01-12-09
Durke

Lesson 8-3  Trigonometry 511

37. a. In △ABC at the right, how does sin A compare to B
16
cos B? Is this true for the acute angles of other 34 C
30
right triangles?

b. Reading Math  The word cosine is derived from the

words complement’s sine. Which angle in △ABC is A
the complement of ∠A? Of ∠B?

c. Explain why the derivation of the word cosine makes sense.

38. For right △ABC with right ∠C, prove each of the following.
Proof a. sin A 6 1
b. cos A 6 1 hsm11gmse_0803_t07653

39. a. Writing  Explain why tan 60° = 13. Include a diagram with your explanation.
b. Make a Conjecture  How are the sine and cosine of a 60° angle related? Explain.

The sine, cosine, and tangent ratios each have a reciprocal A

ratio. The reciprocal ratios are cosecant (csc), secant (sec), 9 15
and cotangent (cot). Use △ABC and the definitions below to

write each ratio.

csc X = sin1 X sec X = 1 X cot X = 1 X C 12 B
cos tan

40. csc A 41. sec A 42. cot A

43. csc B 44. sec B 45. cot B
hsm11gmse_0803_t07654
46. Graphing Calculator  Use the table feature of your graphing calculator to study

sin X as X gets close to (but not equal to) 90. In the y= screen, enter Y1 = sin X.

a. Use the tblset feature so that X starts at 80 and changes by 1. Access the table .
From the table, what is sin X for X = 89?

b. Perform a “numerical zoom-in.” Use the tblset feature, so that X starts with 89
and changes by 0.1. What is sin X for X = 89.9?

c. Continue to zoom-in numerically on values close to 90. What is the greatest

value you can get for sin X on your calculator? How close is X to 90? Does your

result contradict what you are asked to prove in Exercise 38a?
d. Use right triangles to explain the behavior of sin X found above.

47. a. Reasoning Does tan A + tan B = tan (A + B) when A + B 6 90? Explain.
b. Does tan A - tan B = tan (A - B) when A - B 7 0? Use part (a) and indirect

reasoning to explain.

C Challenge Verify that each equation is an identity by showing that each expression on the
left simplifies to 1.

48. (sin A)2 + (cos A)2 = 1 49. (sin B)2 + (cos B)2 = 1 B

50. (cos1A)2 - (tan A)2 = 1 51. 1 - 1 = 1 a c
(sin A)2 (tan A)2

# 52. Show that (tan A)2 - (sin A)2 = (tan A)2 (sin A)2 is Cb A

an identity.

512 Chapter 8  Right Triangles and Trigonometry hsm11gmse_0803_t07655

STEM 53. Astronomy  The Polish astronomer Nicolaus Copernicus devised Outer planet’s orbit
a method for determining the sizes of the orbits of planets farther

from the sun than Earth. His method involved noting the number A Earth’s orbit
of days between the times that a planet was in the positions labeled A Sun
A and B in the diagram. Using this time and the number of days in
1 AU
each planet’s year, he calculated c and d. c˚ d˚
a. For Mars, c = 55.2 and d = 103.8. How far is Mars from the sun
B
in astronomical units (AU)? One astronomical unit is defined as

the average distance from Earth to the center of the sun, about

93 million miles. B Not to scale
b. For Jupiter, c = 21.9 and d = 100.8. How far is Jupiter from the

sun in astronomical units?

PERFORMANCE TASK MATHEMATICAL

Apply What You’ve Learned PRACTICES
MP 1
Look back at the information on page 489 about the fire in a state forest.
The diagram is shown again below.

CN

W E
S
b za

Lookout 54° 30° Lookout
Tower A Tower B
2000 m

Select all of the following that are true. Explain your reasoning.

A. sin 54° = bz
B. cos 30° = 20z00
C. tan 30° = 20z00
D. sin 30° = az
E. tan 54° = 20z00
F. cos 54° = bz

Lesson 8-3  Trigonometry 513

Concept Byte Complementary CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Angles and
Use With Lesson 8-3 Trigonometric Ratios GM-ASFRST.9C1.72 .GEx-pSlRaTin.3a.n7d  Euxspelathinearenldatuiosensthiep
breeltawtieoennshtihpebseintwe eaenndtchoesisnineeoaf ncdomcopsleinmeeonftary
Activity aconmglpelse.
mentary angles.

MP 3

The acute angles of a right triangle are complementary because the sum of R
their measures is 90. There is a relationship between the sine and cosine of
complementary angles. 30Њ
2 √3

60Њ 1 P
What is the relationship between the sine and cosine of complementary Q
angles? B

1. Refer to right triangle PQR, shown at the right. ca
a. How is the side opposite ∠R related to the side adjacent to ∠Q?
b. How is the side adjacent to ∠R related to the side opposite ∠Q? hsm12gese_cc04_t05001.ai
c. Compare sin R and cos Q.
d. Compare sin Q and cos R. Ab C
e. Make a conjecture based on your results from parts (c) and (d).

2. Refer to right triangle ABC, shown at the right.
a. Write sine and cosine ratios for angles B and C. What do you notice

about the relationship between sin B and cos C? Between sin C and
cos B?
b. If ∠B and ∠C are the complementary angles of a right triangle, is the
statement sin B = cos C always true? Explain.

3. Reasoning  Prove that in a right triangle with acute angles B and C,
cos B = sin (90 - m∠B).

Exercises

4. In the diagram at the right, line c is perpendicular to line d. c
hsmO12gese_cc04_t050P02.ai
a. In △LMN, cos L is equal to the sine of which angle?
b. In △MOP, sin P is equal to the cosine of which angle? RQ
c. In △MRQ, cos Q is equal to the sine of which angle? M

5. Reasoning  Refer to the diagram for Exercise 4. Suppose you also know that L
OP } RQ. For which angles in the diagram is the cosine of the angle equivalent N
to the sine of ∠POM? Explain. d

6. A taut wire runs from the top of a tall pole to the ground. The pole is hsm12gese_cc04_t05003.ai

perpendicular to the ground and the ground is level. The sine of the angle that
12
the wire makes with the ground is 13 . What is the cosine of the angle that the

wire makes with the top of the pole? Explain.

514 Concept Byte  Complementary Angles and Trigonometric Ratios

8 Mid-Chapter Quiz MathX

OLMathXL® for School
R SCHO Go to PowerGeometry.com
L®

FO

Do you know HOW? Does each set of numbers form a Pythagorean triple?
Explain.
Algebra  Find the value of each variable. Express your
answer in simplest radical form. 13. 32, 60, 68

1. 15 2. 14. 1, 2, 3
x xy 15. 2.5, 6, 6.5

9 16. Landscaping  A landscaper uses a 13-ft wire to brace

10 a tree. The wire is attached to a protective collar

3. 4. around the trunk of the tree. If the wire makes a
60° angle with the ground, how far up the tree is the
h12sm11gmsye_08mq_t08542 a
protective collar located? Round to the nearest
hs4m5Њ11gmse_08mq_t08545 tenth of a foot.
9 V2
17 Algebra  Find the value of x. Round to the nearest

5. 2hVsm3 11gmxse30y_Њ08mq_t068.5 4h7sm111g8mse_08mx q_t0854 t81e7n.t h 7. x 18. 64
25Њ
31
xЊ

15

Given the following triangle side lengths, identify the 19. 100 20. xЊ

trianghlesmas1a1cgutme,sreig_h0t8, omr qob_ttu0s8e5. 50 hxsm11gms1e2_Њ08mq_t08571.ah4ism11gm7 se_08mq_t08572.ai
7. 7, 8, 9
hsm11gmse_08mq_t08567

8. 15, 36, 39 Do you UNDERSTAND?

9. 10, 12, 16 21. Compare and Contrast  What are the similarities
bhestmwe1e1ngmthseem_0et8hmodqs_ty0o8u5u7s3e.ahtoismde1te1rgmmisnee_w0h8emthqe_rt08574.ai
10. A square has a 40-cm diagonal. How long is each a triangle is acute, obtuse, or right? What are
side of the square? Round to the nearest tenth of the differences?
a centimeter.
22. In the figure below, which angle has the greater sine
Write the sine, cosine, and tangent ratios for value? The greater cosine value? Explain.
jA and jB.
1
11. A 12. A 72 C 2
30
4 6.4 78 23. Reasoning  What angle has a tangent of 1? Explain.
B (Do not use a calculator or a table.)

C5B hsm11gmse_08mq_t08575.ai

hsm11gmse_08mq_t08570.ai

hsm11gmse_08mq_t08569.ai Chapter 8  Mid-Chapter Quiz 515

8-4 Angles of Elevation CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
and Depression
MG-ASFRST..9C1.82 .GU-sSeRtTri.g3o.8n oUmseetrtircigroantiooms e. t.r.ictorastoiolvse. . .
rtioghsotltvreiarnighletstriniaanpgpleliseidnparpopblliemd sp.roblems.

MP 1, MP 3, MP 4, MP 6

Objective To use angles of elevation and depression to solve problems

Did you know you You are on the lighting crew for the school musical. You hang a set of
could use geometry lights 25 ft above the stage. For one song, the female lead is on stage
in theater? You alone and you want all the lights on her. If she stands in the middle
can find math of the stage as shown, at what angle from horizontal should you set
anywhere you lamps A and B? Round to the nearest degree. Describe how each angle
look . . . up or changes if you set the lamps for her to stand a few feet closer to the
down. tree. (Diagram is not to scale.)

AB

MATHEMATICAL

PRACTICES

10 ft 10 ft

Lesson The angles in the Solve It are formed below the horizontal black pipe. Angles formed
above and below a horizontal line have specific names.
Vocabulary
• angle of Suppose a person on the ground sees a hang glider at a
388 angle above a horizontal line. This angle is the angle of
elevation elevation.
• angle of
At the same time, a person in the hang glider sees the person
depression on the ground at a 388 angle below a horizontal line. This
angle is the angle of depression.
Horizontal line

Angle of 38
depression

Notice that the angle of elevation is congruent to the angle of Angle of
depression because they are alternate interior angles. 38 elevation

Essential Understanding  You can use the angles of Horizontal line
elevation and depression as the acute angles of right triangles
formed by a horizontal distance and a vertical height.

516 Chapter 8  Right Triangles and Trigonometry HSM11GMSE_0804_a08342
2nd pass 01-13-09
Durke

Problem 1 Identifying Angles of Elevation and Depression

How can you tell What is a description of the angle as it
if it is an angle relates to the situation shown?
do ef perleevssaitoion?n or
A j1

P lace your finger on the ∠1 is the angle of depression from
vertex of the angle. Trace
along the nonhorizontal the bird to the person in the hot-air
side of the angle. See balloon.

i f your finger is above B j4
((edleepvraetsiosino)no) rthbeelvoewrtex. ∠4 is the angle of elevation from
the base of the mountain to the

person in the hot-air balloon.

Got It? 1. Use the diagram in Problem 1. What is a description of the angle as it relates to

the situation shown?
a. ∠2 b. ∠3

Problem 2 Using the Angle of Elevation

Wind Farm  Suppose you stand 53 ft from a wind
farm turbine. Your angle of elevation to the hub of
the turbine is 56.58. Your eye level is 5.5 ft above
the ground. Approximately how tall is the
turbine from the ground to its hub?

tan 56.5° = 5x3 Use the tangent ratio.

x = 53(tan 56.5°) Solve for x.

53 tan 56.5 enter Use a calculator. x ft

Why does your eye 80.07426526
level matter here?
Your normal line of sight So x ≈ 80, which is the height from your 56.5º
is a horizontal line. The eye level to the hub of the turbine.
angle of elevation starts 53 ft
from this eye level, not To find the total height of the
from the ground.
turbine, add the height from

the ground to your eyes.
Since 80 + 5.5 = 85.5,
the wind turbine is

about 85.5 ft tall from

the ground to its hub.

Got It? 2. You sight a rock climber on a cliff at a 328 angle of Eye level Climber
elevation. Your eye level is 6 ft above the ground and
you are 1000 ft from the base of the cliff. What is the 32Њ
approximate height of the rock climber from the ground? 1000 ft

Lesson 8-4  Angles of Elevation and Depression 517

hsm11gmse_0804_t07812.ai

Problem 3 Using the Angle of Depression 3Њ Angle of descent

To approach runway 17 of the Ponca City Municipal Airport
Airport in Oklahoma, the pilot must begin a 38 descent Not to scale
starting from a height of 2714 ft above sea level. The
airport is 1007 ft above sea level. To the nearest tenth
of a mile, how far from the runway is the airplane at
the start of this approach?

The airplane is 2714 - 1007, or 1707 ft, above the level of the airport.

Why is the angle of 3Њ x hsm11gmse_0804_t07813.ai
elevation also 3°? 3Њ
The path of the airplane 1707 ft
before descent is parallel
to the ground. So the sin 3° = 1707 Use the sine ratio. 6.2
angles formed by the x
path of descent are
congruent alternate x = s1i7n037° Solve for x. 00000 0
interior angles. hsm11gmse_0804_t07815.ai 11111 1
1707 sin 3 enter 32616.19969 Use a calculator. 22222 2
33333 3
5280 enter 6.177310548 Divide by 5280 to convert feet to miles. 44444 4
55555 5
The airplane is about 6.2 mi from the runway. 66666 6
77777 7
88888 8
99999 9

Got It? 3. An airplane pilot sights a life raft at a 268 angle of depression. The
airplane’s altitude is 3 km. What is the airplane’s horizontal distance d from
the raft?

hsm11gmse_0804b_t07665.a

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

What is a description of each angle as it relates to 7. Vobabulary  How is an angle of elevation formed?

the diagram? B 8. Error Analysis  A homework question says that the
1. ∠1 5 angle of depression from the bottom of a house
window to a ball on the ground is 208. Below is your
2. ∠2 D friend’s sketch of the situation. Describe your friend’s
error.
3. ∠3 43 A
4. ∠4 2
5. ∠5
1
C

6. What are two pairs of congruent angles in the 20
diagram above? Explain why they are congruent.

hsm11gmse_0804_t07799

518 Chapter 8  Right Triangles and Trigonometry hsm11gmse_0804_t07821.ai

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Describe each angle as it relates to the situation in the diagram. See Problem 1.

9. ∠1 10. ∠2 11. ∠3 12. ∠4
16. ∠8
13. ∠5 14. ∠6 15. ∠7

4 87
56
3 Max Maya
2
1

Find the value of x. Round to the nearest tenth of a unit. HSM11GMSE_0804_a08349See Problem 2.

17. 22DH100nSuЊ0dMrkfpt1ea1sGsM0S1E-1_3x0-80094_a08348 18. 2nd pass 01-13-09

Durke 203 m

22Њ
x

STEM 19. Meteorology  A meteorologist measures the angle of elevation of a weather
balloon as 418. A radio signal from the balloon indicates that it is 1503 m from his

lhoscmati1o1ng. Tmosthee_0n8ea0r4es_ttm07e8te2r,5h.aoiw high above the grouhnsmd i1s 1thgembaslelo_o0n8?04_t07827.ai

Find the value of x. Round to the nearest tenth of a unit. See Problem 3.

20. 27Њ x 21. 18Њ
2 km
580 yd x

22. Indirect Measurement  A tourist looks out from the crown of the Statue of Liberty,
approximately 250 ft above ground. The tourist sees a ship coming into the harbor

satnahdtsumme et1oa1stguhremessshteihp_e0tao8nt0gh4lee_notef0ad7ree8ps3tre0fos.saoiito.n as 188. Find the dhissmtan1c1egfrmomset_h0e8b0a4se_ot0f 7th8e33.ai

B Apply 23. Flagpole  The world’s tallest unsupported flagpole is a 282-ft-tall steel pole in
Surrey, British Columbia. The shortest shadow cast by the pole during the year is
137 ft long. To the nearest degree, what is the angle of elevation of the sun when
casting the flagpole’s shortest shadow?

Lesson 8-4  Angles of Elevation and Depression 519

24. Think About a Plan  Two office buildings are 51 m apart. The height of the taller
building is 207 m. The angle of depression from the top of the taller building to
the top of the shorter building is 158. Find the height of the shorter building to the
nearest meter.

• How can a diagram help you?
• How does the angle of depression from the top of the taller building relate to the

angle of elevation from the top of the shorter building?

Algebra  The angle of elevation e from A to B and the angle of depression d from B
to A are given. Find the measure of each angle.

25. e: (7x - 5)°, d: 4(x + 7)° 26. e: (3x + 1)°, d: 2(x + 8)°

27. e: (x + 21)°, d: 3(x + 3)° 28. e: 5(x - 2)°, d: (x + 14)°

29. Writing  A communications tower is located on a plot Tower
of flat land. The tower is supported by several guy wires.
Assume that you are able to measure distances along the Guy
ground, as well as angles formed by the guy wires and wires
the ground. Explain how you could estimate each of the
following measurements.

a. the length of any guy wire
b. how high on the tower each wire is attached

Flying  An airplane at a constant altitude a flies a horizontal distance d toward you

at velocity v. You observe for time t and measure its angles of elevation jE1 and
jE2 at the start and end of your observation. Find the missing information.
30. a = ■ mi, v = 5 mi>min, t =
1 min, m∠E1 = 45, m∠E2 = 90 hsm11gmse_0804_t07835.ai

31. a = 2 mi, v = ■ mi>min, t = 15 s, m∠E1 = 40, m∠E2 = 50

32. a = 4 mi, d = 3 mi, v = 6 mi>min, t = ■ min, m∠E1 = 50, m∠E2 = ■

33. Aerial Television  A blimp provides aerial television views of a football game. The
television camera sights the stadium at a 78 angle of depression. The altitude of
the blimp is 400 m. What is the line-of-sight distance from the television camera
to the base of the stadium? Round to the nearest hundred meters.

YCouhraNnewnseLlea1d2er Not to scale

7Њ

400 m

HSM11GMSE_0804_a08350
2nd pass 01-13-09
Durke

520 Chapter 8  Right Triangles and Trigonometry

C Challenge 34. Firefighting  A firefighter on the ground sees fire break through a
window near the top of the building. The angle of elevation to the
windowsill is 288. The angle of elevation to the top of the building
is 428. The firefighter is 75 ft from the building and her eyes are 5 ft
above the ground. What roof-to-windowsill distance can she
report by radio to firefighters on the roof?

35. Geography  For locations in the United States, the relationship 42؇
28؇
between the latitude / and the greatest angle of elevation a of the
sun at noon on the first day of summer is a = 90° - / + 23.5°. 75 ft
Find the latitude of your town. Then determine the greatest angle

of elevation of the sun for your town on the first day of summer.

Not to scale

Standardized Test Prep

SAT/ACT 36. A 107-ft-tall building casts a shadow of 90 ft. To the nearest whole degree, what is
the angle of elevation of the sun?

338 408 508 578

37. Which assumption should you make to prove indirectly that the sum of the
measures of the angles of a parallelogram is 360?

The sum of the measures of the angles of a parallelogram is 360.
The sum of the measures of the angles of a parallelogram is not 360.
The sum of the measures of consecutive angles of a parallelogram is 180.
The sum of the measures of the angles of a parallelogram is 180.

Extended 38. A parallelogram has four congruent sides.
Response a. Name the types of parallelograms that have this property.

b. What is the most precise name for the figure, based only on the given

description? Explain.

c. Draw a diagram to show the categorization of parallelograms.

Mixed Review

Find the value of x. Round to the nearest tenth of a unit. See Lesson 8-3.
6 in.
39. 40. 24Њ 41.
40 m x 94 ft 6 in.
28Њ x
xЊ

Get Ready!  To prepare for Lesson 8-5, do Exercises 42–44.

F4i2n.d (t0h, e0)dhaissnmtdan1(8c1,eg2b)m etswee_e0n8e0a4c_htp0a7i84r 3o3.f6 p(.ah-oisi1nm5t,s1.-12g)masned_(008,004) _t07810.ai44. hsm11gmse_0S8e0e4L_et0ss7o8n111.-a7i.
( -2, 12) and (0, 0)

Lesson 8-4  Angles of Elevation and Depression 521

8-5 Law of Sines MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Objectives To apply the Law of Sines GM-ASFRST..9D1.121.G  -USnRdTe.4rs.t1a1n dUanndderasptapnlydtahnedLapwployfthe
SLainweso.f .S.intoesfi.n.d. tuonfkindowunkmnoewasnumreemaesnutrseminernitgshitn
aringdhtnaonnd-rnigohnt-rtirgiahnt gtrlieasn.g.le.sA.l.s.oAGls-oSRT.D.10
MMAPF1S,.M91P2.3G,-MSRPT.44,.1M0P 7
MP 1, MP 3, MP 4, MP 7

A rescue boat spots a lost hiker on the edge of a
rock shelf. How far is the boat from the hiker?

What additional x
line can you draw
to help solve the 66 ft
problem?
35° 135 ft 68°

MATHEMATICAL

PRACTICES
In the Solve It, you used what you know about triangles to find missing lengths.

Lesson Essential Understanding  If you know the measures of two angles and the
length of a side (AAS or ASA), or two side lengths and the measure of a nonincluded
Vocabulary obtuse angle (SSA), then you can find all the other measures of the triangle.
• Law of Sines

Key Concept  Law of Sines

For any △ABC, let the lengths of the sides opposite angles A, B, and C be a, b, and c, respectively.
Then the Law of Sines relates the sine of each angle to the length of the opposite side.

sin A = sin B = sin C C
a b c b
a

AcB

Here’s Why It Works  Draw the altitude from C to AB and label C

it h. △ACD and △BCD asarinesirnBigB=htha=t rhia ngDMleeusfli.tniiptliiocnatoiofnsiPnreopehrtysmof E1q1ugalmityse_A080b4bhD_t0765a8.ai

sin A = h and B
b and

b sin A = h

b sin A = a sin B Transitive Property of Equality

sin A = sin B Division Property of Equality geom12_se_ccs_C08_L05_t0001.ai
a b

522 Chapter 8  Right Triangles and Trigonometry

Problem 1 Using the Law of Sines (AAS)

In △ABC, m∠A = 48, m∠B = 93, and AC = 15. To the nearest tenth, what is the
length of BC?

How will drawing Draw and label △ABC. You are C
a diagram help you given two angle measures and the 15
solve the problem? length of a nonincluded side (AAS).
Drawing a diagram will 93Њ 48Њ A
help you to visualize the Use the Law of Sines to write an
problem. Carefully draw equation. B
a diagram and label it
with all of the given Solve for BC. sin 93° = sin 48°
information. 15 BC
Use a calculator to find BC.
  gBeCo=m115s2isn_ins9e34_°8c°cs_C08_L05_t0002.ai

BC ≈ 11.16247016

The length of BC is about 11.2.
Got It? 1. In △ABC above, what is AB to the nearest tenth?

You can also use the Law of Sines to find missing angle measures.

Problem 2 Using the Law of Sines (SSA)

In △RST , RT = 11, ST = 18, and mjR = 120. To the nearest tenth, what is mjS?

What unknown angle Step 1 Draw and label a diagram.
should you use?
Use ∠S because it is Step 2 Use the Law of Sines to set up an equation. R
opposite a known side 120Њ 11
sin 120° = sin S
length. 18 11 S 18 T

Step 3 Find m∠S.

sin S = 11 sin 120° Solve for sin S.
18

m∠S = sin-1a 11 sin 120° b ≈ 31.95396690 Use theginevoermse.12_se_ccs_C08_L05_t0003.a
18

m∠S is about 32.0.

Got It? 2. In △KLM, LM = 9, KM = 14, and m∠L = 105. To the nearest tenth, what is
m∠K ?

Lesson 8-5  Law of Sines 523

You can apply the Law of Sines to real-world problems involving triangles.

Problem 3 Using the Law of Sines to Solve a Problem

A ship has been at sea longer than expected and has Port City ship
only enough fuel to safely sail another 42 miles. Port Lighthouse
City Lighthouse and Cove Town Lighthouse are located
40 miles apart along the coast. At sea, the captain cannot 70Њ
determine distances by observation. The triangle formed
by the lighthouses and the ship is shown. Can the ship sail 40 mi
safely to either lighthouse?
What other
information do you Step 1 Find the measure of the angle formed by Port City, 52Њ
need to know to use the cruise ship, and Cove Town.
the Law of Sines? Cove Town
You can use the Law The sum of the measures of the angles of a triangle Lighthouse
of Sines to find the is 180°. Subtract the given angle measures from 180.
distances from the ship
to each lighthouse if you 180 - 70 - 52 = 58
can find the angle with
the ship as its vertex. Step 2 Use the Law of Sines to find the distances from the shgipetoomea1c2h_lsigeh_tchcosu_seC.08_L05_t0004.ai

Port City Cove Town

# ##sin 58° = sin 52° sin 58° = sin 70°
40 d 40 d
# ##Law of Sines   

d sin 58° = 40 sin 52° Find the cross products. d sin 58° = 40 sin 70°

d = 40sins5in8°52° Divide each side by sin 58°.      d = 40 sin 70°
sin 58°
d ≈ 37.16821049 d ≈ 44.32260977
Use a calculator.  

The distance from the ship to Port City Lighthouse is about 37.2 miles, and the distance
to Cove Town Lighthouse is about 44.3 miles.

The ship can sail safely to Port City Lighthouse but not to 2nd Right-fielder
Cove Town Lighthouse. Base
68Њ
Got It? 3. The right-fielder fields a softball between first base
and second base as shown in the figure. If the 60 ft 40Њ
right-fielder throws the ball to second base, how 1st
far does she throw the ball? Base

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
1. In △ABC, AB = 7, BC = 10, and m∠A = 80. To the
PRACTICES
nearest tenth, what is m∠C?
4. Reasoning  If you know thgeetohmree1s2i_dseele_ncgctsh_sCo0f 8_L05_t0005.ai

a triangle, can you use the Law of Sines to find the

2. What is x? G 30Њ missing angle measures? Explain.
3. What is y? 16
K 5. Error Analysis  In △PQR, PQ = 4 cm, QR = 3 cm,
y and m∠R = 75. Your friend uses the Law of Sines to
x sin 75° sin P
62Њ H write 3 = 4 to find m∠P. Explain the error.

524 Chapter 8  Right Triangles and Trigonometry

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Use the information given to solve. See Problems 1 and 2.

6. In △ABC, m∠A = 70, m∠C = 62, and BC = 7.3. To the nearest tenth,
what is AB?

7. In △XYZ, m∠Y = 80, XY = 14, and XZ = 17. To the nearest tenth, what is m∠Z?

Use the Law of Sines to find the values of x and y. Round to the nearest tenth.

8. 9. 5 10. 11. 14
12 41Њ 62Њ
x 18 x 119Њ 22Њ xЊ 18
y 38Њ xy

63Њ 71Њ y
y

12. The main sail of a sailboahtshmas1t1hge mse_0804b_4t20Њ7661.ai See Problem 3.

dimensions shown in the figure at the h hsm11gmse_0804b_t07662h.asim11gmse_0804b_t0766

rhigshmt. 1To1gthme nseea_r0e8st0t4enbt_hto0f7a6f5o9ot.a, wi hat
is the height of the main sail?
48Њ 54Њ Maple St
B Apply 13. A portion of a city map is shown in the 12 ft

figure at the right. If you walk along Maple 210 yd 88Њ
Street between 2nd Street and Elm Grove Lane, how far do
you walk? Round your answer to the nearest tenth of a yard. Elm Grove 2nd St
Lane

14. Navigation  The Bermuda Triangle is a hgisetoormica1l2ly_fsaem_ocucss_C08_L05_t0007 .ai Bermuda

region of the Atlantic Ocean. The vertices of the triangle are 62Њ

formed by Miami, FL; Bermuda; and San Juan, Puerto Rico. Miami

The approximate dimensions of the Bermuda Triangle are 960 mi

shown in the figure at the right. Explain how you would find geom12_se_ccs_C08_L05_t0008 .ai

the distance from Bermuda to Miami. What is this distance 63Њ
to the nearest mile? San Juan

15. Think About a Plan  An airplane took off from an airport and started flying toward

its destination 210 miles due east. After flying 80 miles east, it encountered a storm

and altered its course by turning left 22°. When it was past the storm, it turned

right 30° and flew in a straight line until it reached its destination. How far was the

plane from its destination when it made the 30° turn?

• Would drawing a diagram help you visualize the problem situatiogne?om12_se_ccs_C08_L05_t0009 .ai

• What are you being asked to find?
• What measures do you know?

Lesson 8-5  Law of Sines 525

16. Zipline  A zipline is constructed over a ravine as 45 ft platform 250 ft 60 ft
shown in the diagram at the right. What is the x 50Њ ladder
horizontal distance from the bottom of the
ladder to the platform where the zipline ends? ground
Round your answer to the nearest tenth of a foot.

17. If m∠DEG = m∠D + m∠G + 43,
what is m∠EFG?
E

8 in. 7.5 in. 8 in.

DF G C

C Challenge 18. You can use the formula Area = 1 bc sin A to find ftohremaurelgaaeom12_se_cbcs_C08_L05_pplr_art.ai
2 how this
of the triangle shown at the right. Show

becomes the more familiar formula for the area of a triangle Ac B
if △ABC is a right triangle and m∠A = 90.

geom12_se_ccs_C08_L05_t0012.ai

PERFORMANCE TASK

Apply What You’ve Learned geom1PM2RA_TAHsCEeMT_AIcTCIcCEAsSL_C08_L05_t0011.ai

MP 6

Look back at the information on page 489 about the fire in a state forest.

The diagram is shown again below.

CN

b za WE
S

Lookout 54° 30° Lookout
Tower A Tower B
2000 m

a. What is the measure of the angle between the distances labeled a and b?

b. Write and solve an equation using the Law of Sines to find the distance between
Lookout Tower A and the fire. Round to the nearest tenth.

c. Write and solve an equation using the Law of Sines to find the distance between
Lookout Tower B and the fire. Round to the nearest tenth.

526 Chapter 8  Right Triangles and Trigonometry

8-6 Law of Cosines CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Objective To apply the Law of Cosines GM-ASFRST.9D1.121.G  -USnRdTe.r4s.t1an1d  Uandearpstpalnydthaend. .a.pLpalwy of
tChoesi.n.e.sL.a.w. AoflsCoosGin-SesR.T.D. A.1l0so
MMAPF1S,.M91P2.3G, -MSRPT4.4, .M10P 7
MP 1, MP 3, MP 4, MP 7

Think about how In the diagram, △ABC is an acute triangle. A
the areas of Use what you know about right triangle cb
the squares are trigonometry to write an expression for B aC
related to the the area of the shaded region that uses
side lengths of a, b, and C.
the triangle. What
does this remind
you of?

MATHEMATICAL In the Solve It, you used right triangle trigonometry to write an expression to describe
PRACTICES a side length. You can also find relationships between the angle measures and the side

lengths of nonright triangles.

Essential Understanding  If you know the measures of two side lengths and the
measure of the included angle (SAS), or all three side lengths (SSS), then you can find
all the other measures of the triangle.

geom12_se_ccs_C08_L06_t0001.ai

Key Concept  Law of Cosines

Lesson For any △ABC, the Law of Cosines relates the cosine of each angle to the side lengths

Vocabulary of the triangle. C
• Law of Cosines a2 = b2 + c2 - 2bc cos A b
b2 = a2 + c2 - 2ac cos B
c2 = a2 + b2 - 2ab cos C a

AcB

Here’s Why It Works  Note that b2 = x2 + h2 and x = b cos A. C

Use the Pythagorean Theorem with △BCD and simplify.hsm11gmse_0b804bh_t0765a8.ai
a2 = (c - x)2 + h2 Pythagorean Theorem

a2 = c2 - 2cx + x2 + h2 Simplify. A x cϪx B
a2 = c2 - 2cb cos A + b2 Substitute b2 for x2 + h2 and b cos A for x. Dc

a2 = b2 + c2 - 2bc cos A Commutative Property

Lesson 8-6  Law of Cosines geom12_se_ccs_C08_L0562_7t0002

Problem 1 Using the Law of Cosines (SAS) A
Find b to the nearest tenth. 10 b
B 44Њ 22
AB is opposite ∠C, the length Because you know m∠B and need C
so AB = c = 10. of AC b, substitute the angle measure
BC is opposite ∠A,
so BC = a = 22. and the two side lengths into
m∠B = 44 b2 = a2 + c2 - 2ac cos B and
solve for b.

b2 = a2 + c 2 - 2ac cos B Law of Cosines hsm11gmse_0804b_t07673.ai

b2 = 222 + 102 - 2(22)(10) cos 44° Substitute. M
N
b ≈ 16.35513644 Use a calculator.
48 104Њ 29
The value of b is about 16.4. L

Got It? 1. Find MN to the nearest tenth.

How can you use Problem 2 Using the Law of Cosines (SSS)
what you know to
find mjV? In △TUV , TU = 4.4, UV = 7.1, and TV = 6.7. Find mjV to the nearesgt eteonmth12_se_ccs_C08_L06_t0003.ai
You know the three side
of a degree.
lengths (SSS) so you can
Step 1 Draw and label a diagram. T 6.7
use the Law of Cosines to Step 2 Use the Law of Cosines to set up an equation. 4.4
find m∠V . TU 2 = UV 2 + TV 2 - 2(UV)(TV) cos V

4.42 = 7.12 + 6.72 - 2(7.1)(6.7) cos V U 7.1 V

Step 3 Solve for m∠V . geom12_se_ccs_C08_L06_t0004.ai

19.36 = 50.41 + 44.89 - 95.14 cos V Simplify.

- 75.94 = cos V Solve for cos V.
- 95.14

V = cos-1a - 75.94 b Solve for m∠V .
- 95.14

m∠V ≈ 37.04219062 Use a caclulator.

The measure of ∠V is about 37.0.

Got It? 2. In △TUV above, find m∠T to the nearest tenth degree.

528 Chapter 8  Right Triangles and Trigonometry

You can use the Law of Cosines to solve real world problems involving triangles.

Problem 3 Using the Law of Cosines to Solve a Problem

An air traffic controller is tracking a plane 2.1 kilometers North
due south of the radar tower. A second plane is located
What do you need 3.5 kilometers from the tower at a heading of N 75° E 3.5 km Plane B
to find before you (75° east of north). To the nearest tenth of a kilometer, Tower 75Њ
can use the law of how far apart are the two planes?
Cosines? d

You need to find the The north-south line in the figure represents a straight angle. Let the 2.1 km
angle opposite d be ∠D. Use supplementary angles to find m∠D. Use
measure of the angle Plane A
supplementary angles to find the measure of the angle opposite d.
opposite d in the triangle
m∠D = 180 - 75 = 105 Supplementary angles
before you can apply the

Law of Cosines.

Use the Law of Cosines to solve for d.

d 2 = a2 + b2 - 2ab cos D Law of Cosines

d 2 = 3.52 + 2.12 - 2(3.5)(2.1) cos 105° Substitute. geom12_se_ccs_C08_L06_t000

d ≈ 4.52378602 Use a calculator.

The distance between the two planes is about 4.5 kilometers.

Got It? 3. You and a friend hike 1.4 miles due west from a campsite. At the same time,
two other friends hike 1.9 miles at a heading of S 11° W (11° west of south)

from the campsite. To the nearest tenth of a mile, how far apart are the

two groups?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

1. In △ABC, AB = 7, BC = 10, and m∠B = 80. PRACTICES
To the nearest tenth, what is b?
5. Error Analysis  In △ABC, AC = 15 ft, BC = 12 ft,
2. In △QRS, QR = 31.9, RS = 25.2, and QS = 37.6. and m∠C = 32. A student solved for c for a = 12 ft,
To the nearest tenth, what is m∠R? b = 15 ft, and m∠C = 32. What was the error?

3. In △LMN , LN = 7, MN = 10, and m∠N = 48. C = 122 + 152 - 2(12)(15)cos32°
To the nearest tenth, what is the area of △LMN ? C = 369 - 360 cos32°
C = 63.7
4. What are m∠X , m∠Y , X 6m
and m∠Z? 4m 6. Reasoning  Explain how you would find the measure
of the largest angle of a triangle if given the measures
Y 7m Z of the three side lengths.

geom12_se_ccs_C08_L06_t0007.ai

geom12_se_ccs_C08_L0L6es_sto0n080-66 .aiLaw of Cosines 529

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Use the information given to solve. See Problems 1 and 2.

7. In △QRS, m∠R = 38, QR = 11, and RS = 16. To the nearest tenth, what is the
length of QS?

8. In △WXY , WX = 20.4, XY = 16.4, and WY = 25.3. To the nearest tenth, what is
m∠W ?

9. In △JKL, JK = 2.6, KL = 6.4, and m∠K = 10.5. To the nearest tenth, what is the
length of JL?

10. In △DEF , DE = 13, EF = 24, and FD = 27. To the nearest tenth, what is m∠E?

Use the Law of Cosines to find the values of x and y. Round to the nearest tenth.

11. 12. yЊ 13. 80 14.
11 14 53
xЊ yЊ xЊ x 40Њ 9 x
19 78
27Њ yЊ
4 yЊ 5

Use the Law of Cosines to solve each problem. See Problem 3.

15. hBishaolssmoemceba1atpe1lldla gt1Aem1fat0sesefrse_fheieo0tlwf8dr0oninm4ignbafti_hrgtser0tohfb7uigas6nusm7der5e1ba.ana1atldiglt,hm5a7epsfrieeitgec_hth0tfe.8rro0m4b_t0h7s6m691.p1aitigchmerse_080141b0 f_tt0h7s6m721.1aigB1msatssee_0804b_t07670.ai
57 ft 90 ft
To the nearest tenth, what is the measure of the angle

with its vertex at the pitcher?

16. Zipline  One side of a ravine is 14 ft long. The other side Home
is 12 ft long. A 20 ft zipline runs from the top of one side Plate

of the ravine to the other. To the nearest tenth, at what

angle do the sides of the ravine meet?

20 ft

12 ft geom12_se_ccs_C08_L06_t0008.ai
14 ft

B Apply 17. Think About a Plan  A walking path around the outside of a garden is shaped like
a triangle. Two sides of the path that measure 32 ft and 39 ft form a 76° angle. If you

walk around the entire path one time, how far have you walked? Write your answer

to the nearest foot.

• Whatgienfoomrm1a2ti_osned_occyso_uCn0e8ed_Lto0f6in_dt0b0e0fo9r.eayiou can solve this problem?

• How can you find the information you need?

• Can drawing a diagram help you solve this problem?

530 Chapter 8  Right Triangles and Trigonometry

18. Airplane  A commuter plane flies from City A to City B, 57 mi City B
a distance of 90 mi due north. Due to bad weather, the plane
is redirected at take-off to a heading N 60° W (60° west of north). 90 mi
After flying 57 mi, the plane is directed to turn northeast and 60Њ
fly directly toward City B. To the nearest tenth, how many miles City A
did the plane fly on the last leg of the trip?

For each triangle shown below, determine whether you would use the Law
of Sines or Law of Cosines to find the value of x. Then find the value of x to
the nearest tenth.

19. 20. 7
86 x

40Њ xЊ 48Њ

10 geom12_se_ccs_C08_L06_t0010

21. 22.
x 40Њ
hsm1127gmse_0804x b_t07677.ai
hsm11gmse_0804b_t07676.ai
36Њ
110Њ 23
12

23. A 15-ft water slide has a 9.5-ft ladder which meets the slide 95Њ 15 ft
at a 95° angle. To the nearest tenth, what is the distance
9.5 ft

bhetswme1en1gthme esend_0o8f t0h4ebs_lidt0e7a6n7d8th.aeibottom of the laddhesr?m11gmse_0804b_t07674.ai
24. Flags  The dimensions of a triangular flag are 18 ft by 25 ft by

27 ft. To the nearest tenth, what is the measure of the angle

formed by the two shorter sides?

C Challenge 25. Parallelogram QRST has a perimeter of 62 mm. To the nearest tenth, what is the
length of TR?

QR geom12_se_ccs_C08_L06_t0011.ai

2x 13.3 m
120Њ 26Њ
T 3x Ϫ 3 S
11Њ 12.4 m
STEM 26. Surveying  A surveyor measures the distance to the base
of a monument to be 12.4 meters at an angle of elevation of
11°. At an angle of elevation of 26°, the distance to the top
of the monument is 13.3 meters. What is the height of the

monugmeeonmt t1o2th_esen_ecarcess_tCte0n8th_?L06_t0013.ai

Lesson 8-6  Law of Cosgineeos m12_se_ccs_C08_L06_pplr_ar5t.3a1i

27. An isosceles triangle XYZ has a base of 12 in. and a height of 8 in. To the nearest
tenth, what are the measures of the angles?

28. Open-Ended  Describe a situation in which you are given three measures of a
triangle but are unable to solve the triangle for the other three measures.

PERFORMANCE TASK MATHEMATICAL

Apply What You’ve Learned PRACTICES
MP 1, MP 2
Look back at the information on page 489 about the fire in a state forest.
The diagram is shown again below.

CN

b za WE
S

Lookout 54° 30° Lookout
Tower A Tower B
2000 m

For parts (a) and (b), refer to the distances you found in the Apply What You
Learned section in Lesson 8-5.

a. Use the Law of Cosines to verify the distance between Lookout Tower A
and the fire.

b. Use the Law of Cosines to verify the distance between Lookout Tower B
and the fire.

c. Write an equation that can be used to find how far west of Lookout
Tower B the fire is located.

532 Chapter 8  Right Triangles and Trigonometry

8 Pull It All Together

RMANCPERFOE TASKCompleting the Performance Task

To solve these Look back at your results from the Apply What You’ve Learned sections in Lessons 8-3,
problems, you 8-5, and 8-6. Use the work you did to complete the following.
will pull together
many concepts 1. Solve the problem in the Task Description on page 489 by finding how far to the
and skills that north and west of Lookout Tower B the fire is located. Round your answers to the
you have studied nearest tenth of a meter. Show all your work and explain each step of your solution.
about right
triangles and 2. Reflect  Choose one of the Mathematical Practices below and explain how you
trigonometry. applied it in your work on the Performance Task.
MP 1: Make sense of problems and persevere in solving them.

MP 2: Reason abstractly and quantitatively.

MP 6: Attend to precision.

On Your Own

Another fire is sighted by the rangers at the two lookout towers. The ranger at Lookout
Tower A observes the smoke at an angle of 65°. The ranger at Lookout Tower B observes
the smoke at an angle of 45°. Both angles are measured from the line that connects the
two lookout towers.

Fire

N

WE

S

Tower A 65° 2000 m 45° Tower B

How far north of Lookout Tower A is this fire located? Round your answer to the nearest
tenth of a meter.

Chapter 8  Pull It All Together 533

8 Chapter Review

Connecting and Answering the Essential Questions

The Pythagorean Theorem (Lesson 8-1) Special Triangles (Lesson 8-2)

1 Measurement a2 + b2 = c2 a c c 60Њ a
Use the Pythagorean ac
Theorem or trigonometric
ratios to find a side b 45Њ
length or angle measure ab
of a right triangle. The
Law of Sines and the Law Trigonometry (Lesson 8-3) c = a 12 c = 2a
of Cosines can be used to b = a 13
find missing side lengths
and angle measures of A hsmH1y1pogtemnussee_08cr_t08543.aiAhnsmgle1s1 ogfm Elseev_a0thi8oscnmr_1t10g8m54s4e._a0i 8cr_t08546.a
any triangle. Adjacent

2 Similarity to ЄA and Depression (Lesson 8-4)
A trigonometric ratio C Opposite ЄAB
compares the lengths
of two sides of a right sin A = opposite Angle of elevation
triangle. The ratios hypotenuse Angle of depression
remain constant within
a group of similar right cos A = adjacent Lhasmw1 1ogfm Ssei_n0e8csr_ at0n85d5 1L.aai w of
triangles. Cosines (Lessons 8-5 and 8-6)
hhyspmote1nu1sge mse_08cr_t08549.ai
opposite
tan A = adjacent

sin A = sin B = sin C
a b c

a2 = b2 + c 2 - 2bc cos A
b2 = a2 + c 2 - 2ac cos B
c 2 = a2 + b2 - 2ab cos C

C

ba

AcB

Chapter Vocabulary • Law of Cosines (p. 526) geom12_se_ccs_C08csr_t01.ai
• Law of Sines (p. 522)
• angle of depression (p. 516) • Pythagorean triple (p. 492) • sine (p. 507)
• angle of elevation (p. 516) • tangent (p. 507)
• cosine (p. 507) • trigonometric ratios (p. 507)

Choose the correct term to complete each sentence.
1. ? are equivalent ratios for the corresponding sides of two triangles.
2. A(n) ? is formed by a horizontal line and the line of sight above that line.
3. A set of three nonzero whole numbers that satisfy a2 + b2 = c2 form a(n) ? .
534 Chapter 8  Right Triangles and Trigonometry

8-1  The Pythagorean Theorem and Its Converse

Quick Review Exercises

The Pythagorean Theorem holds true for any right triangle. Find the value of x. If your answer is not an integer,
express it in simplest radical form.
(leg1)2 + (leg2)2 = (hypotenuse)2

a2 + b2 = c2 4. 14 5. x
8
The Converse of the Pythagorean Theorem states that
if a2 + b2 = c2, where c is the greatest side length of a x 16 15
triangle, then the triangle is a right triangle.

Example 6. 7. hsm11gmse_08cr_t07692.ai

What is the value of x? x x9

a2 + b2 = c2 Pythagorean Theorem x 20 hsm11gmse_08cr_t07686.ai
x2 + 122 = 202 Substitute. 12
x2 = 256 Simplify. 12 18
x = 16 Take the square root.

8-2  Special Right Triangles hsm11gmse_08cr_t07694h.asmi 11gmse_08cr_t07696.ai

hsm11gmse_08cr_t07685.ai

Quick Review Exercises

#45° -45° -90° Triangle Find the value of each variable. If your answer is not an
integer, express it in simplest radical form.
hypotenuse = 12 leg

##30° -60° -90° Triangle 8. y 9. 45Њ x
7 10
hypotenuse = 2 shorter leg 45Њ
longer leg = 13 shorter leg x

Example

What is the value of x? 10. 11.

The triangle is a 30° -60° -90° triangle, 20 60Њ hsm310y1Њxgmse_068cr_t07699hy.asim1114gmse_08cr_t07701.ai
and x represents the length of the
30Њ 60Њ
#longer leg. x
x
longer leg = 13 shorter leg

x = 2013 12. Aahhssmqous1ea1rfregogmmarsodene_en0ch8oacrsnrs_eirdt0oes7f t57h00ef2tg.alaorindeg.nYtoouasntorethtcehr
hsm11gmse_08cr_t07697tc.eoanritnhe,rhaolownlgonthgeisgathrdeehno’ssed?iaghosnmal.1T1ogtmhesnee_a0re8sctr_t07705.ai

Chapter 8  Chapter Review 535

8-3 and 8-4  Trigonometry and Angles of Elevation and Depression

Quick Review Exercises

In right △ABC, C is the right angle. A Express sin A, cos A, and tan A as ratios.

sin ∠A = leghoyppoptoesnitues∠e A Adjacent Hypotenuse 13. B 14. A
2 V19 12
cos ∠A = leg adjacent to ∠A to ЄA C 20 20
hypotenuse C Opposite ЄA B 18 A

tan ∠A = lelgegadopjapcoesnitteto∠∠AA C 16 B

Example Find the value of x to the nearest tenth.

hsm11gmse_08cr_t1186 155.a. i hsm11gmse_08cr_t071760. 9.ai

What is FE to the nearest tenth? D F 12 h12sm11gm22se_08cr_t07710.ai

You know the length of the hypotenuse, 36Њ xЊ
and FE is the side adjacent to ∠E. x
9 41Њ

cos 41° = FE Use cosine. E 17. While flying a kite, Linda lets out 45 ft of string and
9
aaohnnfstgcmhlheeo1okr1fsitgeietlmeftrvoosatemthi_oetn0hg8oerocfgutrrh_noetud0k.ni7Sdteh7?ei1Rs1do5he.u8ast°nemi.drWm1to1hinagtehtmseistnshtheeaea_t rt0hehe8seitcgrh_t t07712.ai
FE = 9(cos 41° ) Multiply each side by 9.

FE ≈ 6.8 Use a calculator.

hsm11gmse_08cr_t07t7en0t7h..ai

8-5 and 8-6  Law of Sines and Law of Cosines

Quick Review Exercises

In △ABC, a, b, and c are the lengths of the sides opposite Find the value of x to the nearest tenth.
∠A, ∠B, and ∠C, respectively. The Law of Sines and the
18. R 12 cm 19. Z
Law of Cosines are summarized below. 20 ft
65Њ Q
sin A sin B sin C C 11 ft
a = b = c b 115Њ
15 cm
a2 = b2 + c 2 - 2bc cos A a x Y

b2 = a2 + c 2 - 2ac cos B Ac B X xЊ
c 2 = a2 + b2 - 2ab cos C
P

Example geoGm12_se_ccs_C08csr_t 2001. .Iaanni d△∠DEFFre, ssipdeecstidv,eely, .aTnhdef  are opposite ∠D, ∠E, in.,
side lengths are d = 25
What is GH?
e = 18 in., and f = 20 in. Find the m∠D to the
Use the Law of Sines to find GH.
14.1 cm nearest tenth.

  sin 46° = sin 80° 21. In △LMN , sides /, m, and n are opposite ∠L, ∠M,
GH 14.1
 GH sin 80° = 14.1 sin 46° and ∠N respectively. You know that m = 3 cm,
genonem=ar18e2sct_mtse,ena_tnhcd.cms_∠CL08=c7s2r_° g.tF0ei3on.dmait1h2e _ms∠e_Ncctos_thCe08csr_t04.ai
0.9848 GH = 10.1427 H 80Њ 46Њ I

        GH ≈ 10.3

536 Chapter 8  Right Triangles and Trigonometry