# 8-1 The Pythagorean Theorem and Its Converse Vocabulary

```8-1
The Pythagorean Theorem
and Its Converse
Vocabulary
Review
1. Write the square and the positive square root of each number.
Number
Square
Positive Square Root
9
81
3
1
4
1
16
1
2
Vocabulary Builder
leg
po
ten
us
e
Related Word: hypotenuse
Definition: In a right triangle, the sides that form the right
angle are the legs.
leg
Main Idea: The legs of a right triangle are perpendicular. The hypotenuse is the side
opposite the right angle.
2. Underline the correct word to complete the sentence.
The hypotenuse is the longest / shortest side in a right triangle.
Write T for true or F for false.
F
3. The hypotenuse of a right triangle can be any one of the three sides.
T
4. One leg of the triangle at the right has length 9 cm.
T
5. The hypotenuse of the triangle at the right has length 15 cm.
Chapter 8
202
12 cm
15 cm
9 cm
hy
leg (noun) leg
Theorems 8-1 and 8-2 Pythagorean Theorem and Its Converse
Pythagorean Theorem If a triangle is a right triangle, then the
sum of the squares of the lengths of the legs is equal to the square
of the length of the hypotenuse.
B
c
If nABC is a right triangle, then a2 1 b2 5 c2 .
A
a
C
b
Converse of the Pythagorean Theorem If the sum of the squares
of the lengths of two sides of a triangle is equal to the square of the
length of the third side, then the triangle is a right triangle.
If a2 1 b2 5 c2 , then nABC is a right triangle.
6. Circle the equation that shows the correct relationship among the lengths of the
legs and the hypotenuse of a right triangle.
132 1 52 5 122
52 1 122 5 132
122 1 132 5 52
Underline the correct words to complete each sentence.
7. A triangle with side lengths 3, 4, and 5 is / is not a right triangle because 32 1 42 is
equal / not equal to 52 .
8. A triangle with side lengths 4, 5, and 6 is / is not a right triangle because 42 1 52 is
equal / not equal to 62 .
Problem 1 Finding the Length of the Hypotenuse
Got It? The legs of a right triangle have lengths 10 and 24. What is the length of
the hypotenuse?
9. Label the triangle at the right.
10. Use the justifications below to find the length of the hypotenuse.
a2 1 b2 5 c2
2
10
Pythagorean Theorem
c
24
2
10 1 24 5 c2
Substitute for a and b.
100 1 576 5 c2
Simplify.
676 5 c2
26 5 c
Take the positive square root.
11. The length of the hypotenuse is 26 .
12. One Pythagorean triple is 5, 12, and 13. If you multiply each number by 2, what
numbers result? How do the numbers that result compare to the lengths of the
sides of the triangle in Exercises 9–11?
10, 24, 26. Answers may vary. Sample: The numbers are the same
_______________________________________________________________________
as the lengths of the sides of the triangle in Exercises 9–11.
_______________________________________________________________________
203
Lesson 8-1
Problem 3 Finding Distance
Got It? The size of a computer monitor is the length of its diagonal. You
19 in.
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.
13. Label the diagram of the computer monitor at the right.
11 in.
b in.
14. The equation is solved below. Write a justification for each step.
a2 1 b2 5 c2
Pythagorean Theorem
112 1 b2 5 192
Substitute.
121 1 b2 5 361
Simplify.
121 2 121 1 b2 5 361 2 121
b2 5 240
Subtract 121 from each side.
Simplify.
b 5 "240
Take the positive square root.
b < 15.49193338
Use a calculator.
Problem 4 Identifying a Right Triangle
Got It? A triangle has side lengths 16, 48, and 50. Is the triangle a right
triangle? Explain.
16. Circle the equation you will use to determine whether the triangle is a right triangle.
162 1 482 0 502
162 1 502 0 482
482 1 502 0 162
17. Simplify your equation from Exercise 16.
162 1 482 0 502
256 1 2304 0 2500
2560 u 2500
18. Underline the correct words to complete the sentence.
The equation is true / false , so the triangle is / is not a right triangle.
A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the
equation a 2 1 b 2 5 c 2 . If you multiply each number in a Pythagorean triple by the
same whole number, the three numbers that result also form a Pythagorean triple.
Chapter 8
204
15. To the nearest tenth of an inch, the width of the monitor is 15.5 in.
Theorems 8-3 and 8-4 Pythagorean Inequality Theorems
Theorem 8-3 If the square of the length of the longest side of a triangle is greater than
the sum of the squares of the lengths of the other two sides, then the triangle is obtuse.
Theorem 8-4 If the square of the length of the longest side of a triangle is less than the
sum of the squares of the lengths of the other two sides, then the triangle is acute.
Use the figures at the right. Complete each sentence with acute or obtuse.
19. In nABC, c 2 . a 2 1 b 2 , so nABC is 9.
A
a
C
b
obtuse
S
t
20. In nRST, s 2 , r 2 1 t 2 , so nRST is 9.
B
c
acute
R
r
T
s
Lesson Check • Do you UNDERSTAND?
Error Analysis A triangle has side lengths 16, 34, and 30. Your friend
says it is not a right triangle. Look at your friend’s work and describe
the error.
21. Underline the length that your friend used as the longest side. Circle
the length of the longest side of the triangle.
16
30
162 + 342 =? 302
256 + 1156 =? 900
1412 ≠ 900
34
22. Write the comparison that your friend should have used to determine whether the
triangle is a right triangle.
162 1 302 0 342
23. Describe the error in your friend’s work. Answers may vary. Sample:
My friend used the wrong length for c in the comparison. The comparison
_______________________________________________________________________
should be 162 1 302 0 342 .
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
hypotenuse
leg
Pythagorean Theorem
Pythagorean triple
Rate how well you can use the Pythagorean Theorem and its converse.
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Lesson 8-1
Special Right Triangles
8-2
Vocabulary
Review
1. Circle the segment that is a diagonal of square ABCD.
AB
AC
BC
D
C
A
B
CD
2. Underline the correct word to complete the sentence.
A diagonal is a line segment that joins two sides / vertices of a polygon.
Vocabulary Builder
complement (noun)
KAHM
pluh munt
Nonexample: Two angles whose measures sum to 180 are supplementary.
Complete each statement with the word complement or complementary.
3. If m/A 5 40 and m/B 5 50, the angles are 9.
complementary
4. If m/A 5 30 and m/B 5 60, /B is the 9 of /A.
complement
5. /P and /Q are 9 because the sum of their measures is 90.
complementary
Complete.
6. If /R has a measure of 35, then the complement of /R has a measure of 55 .
7. If /X has a measure of 22, then the complement of /X has a measure of 68 .
8. If /C has a measure of 65, then the complement of /C has a measure of 25 .
9. Circle the complementary angles.
60í
Chapter 8
40í
50í
206
120í
Math Usage: When the measures of two angles have a sum of 90, each angle is a
complement of the other.
Theorem 8-5 45°-45°-90° Triangle Theorem
In a 458-458-908 triangle, both legs are congruent and the length of the
hypotenuse is "2 times the length of a leg.
45
s 62
Complete each statement for a 458 2458 2908 triangle.
10. hypotenuse 5 "2 ? leg
45
s
s
11. If leg 5 10, then hypotenuse 5 "2 ? 10 .
Problem 1 Finding the Length of the Hypotenuse
Got It? What is the length of the hypotenuse of a 458-458-908 triangle
with leg length 5!3 ?
12. Use the justifications to find the length of the hypotenuse.
hypotenuse 5 "2 ? leg
458-458-908 Triangle Theorem
5 "2 ? 5 "3
Substitute.
5 5 "2 ? "3
Commutative Property of Multiplication.
5 5 "6
Simplify.
Problem 2 Finding the Length of a Leg
Got It? The length of the hypotenuse of a 458-458-908 triangle is 10. What is the
length of one leg?
13. Will the length of the leg be greater than or less than 10? Explain.
Less than. Explanations may vary. Sample: The hypotenuse is the longest side.
__________________________________________________________________________________
14. Use the justifications to find the length of one leg.
hypotenuse 5 "2 ? leg
10 5 "2 ? leg
10
"2
5
leg 5
leg 5
leg 5
"2
"2
? leg
10
"2
?
10 "2
2
leg 5 5 "2
Substitute.
Divide each side by "2 .
Simplify.
"2
10
458-458-908 Triangle Theorem
"2
"2
Multiply by a form of 1 to rationalize the denominator.
Simplify.
Divide by 2.
207
Lesson 8-2
Problem 3 Finding Distance
Got It? You plan to build a path along one diagonal of a 100 ft-by100 ft square garden. To the nearest foot, how long will the path be?
15. Use the words path, height, and width to complete the diagram.
height
16. Write L for leg or H for hypotenuse to identify each part of the right
triangle in the diagram.
H
path
L
height
path
L width
width
17. Substitute for hypotenuse and leg. Let h 5 the length of the hypotenuse.
hypotenuse 5 "2 ? leg
h
5 "2 ? 100
18. Solve the equation. Use a calculator to find the length of the path.
h 5 !2 ? 100
h N 141.4213562
19. To the nearest foot, the length of the path will be 141 feet.
Theorem 8-6 30°-60°-90° Triangle Theorem
In a 308-608-908 triangle, the length of the hypotenuse is twice the length of the
shorter leg. The length of the longer leg is "3 times the length of the shorter leg.
2s 30 s V3
20. hypotenuse 5
2
60
s
? shorter leg
21. longer leg 5 "3 ? shorter leg
Problem 4 Using the Length of One Side
Got It? What is the value of f in simplest radical form?
5œ3
3
22. Complete the reasoning model below.
Write
f is the length of the hypotenuse. I can write an
hypotenuse â 2 Ƃ shorter leg
fâ 2 Ƃ
Now I can solve for f.
Chapter 8
fâ
208
10ƃ3
3
60˚
30˚
f
Think
equation relating the hypotenuse and the
5ƃ3
shorter leg
of the 30í-60í-90ítriangle.
3
5
5ƃ3
3
Complete each statement for a 308-608-908 triangle.
Problem 5 Applying the 30°-60°-90° Triangle Theorem
Got It? Jewelry Making An artisan makes pendants in the shape of
equilateral triangles. Suppose the sides of a pendant are 18 mm long.
What is the height of the pendant to the nearest tenth of a millimeter?
18 mm
18 mm
23. Circle the formula you can use to find the height of the pendant.
hypotenuse 5 2 ? shorter leg
longer leg 5 !3 ? shorter leg
18 mm
24. Find the height of the pendant.
longer leg 5 "3 ? shorter leg
5 "3 ? 9
N 15.58845727
25. To the nearest tenth of a millimeter, the height of the pendant is 15.6 mm.
Lesson Check • Do you UNDERSTAND?
Reasoning A test question asks you to find two side lengths of a 45°-45°-90° triangle.
You know that the length of one leg is 6, but you forgot the special formula for
45°-45°-90° triangles. Explain how you can still determine the other side lengths.
What are the other side lengths?
26. Underline the correct word(s) to complete the sentence. In a 45°-45°-90° triangle,
the lengths of the legs are different / the same .
27. Use the Pythagorean Theorem to find the length of the longest side.
longest side: c2 5 62 1 62
c2 5 36 1 36
c2 5 72
c 5 "72 5 6 "2
28. The other two side lengths are 6 and 6 "2 .
Math Success
Check off the vocabulary words that you understand.
leg
hypotenuse
right triangle
Pythagorean Theorem
Rate how well you can use the properties of special right triangles.
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Lesson 8-2
Trigonometry
8-3
Vocabulary
Review
The Venn diagram at the right shows the relationship between similar and
congruent figures. Write T for true or F for false.
F
T
T
1. All similar figures are congruent figures.
Similar Figures
Congruent
Figures
2. All congruent figures are similar figures.
3. Some similar figures are congruent figures.
4. Circle the postulate or theorem you can use to verify that the triangles at the
right are similar.
AA , Postulate
SAS , Theorem
SSS , Theorem
ratio (noun)
RAY
shee oh
Related Words: rate, rational
Definition: A ratio is the comparison of two quantities by division.
6
Example: If there are 6 triangles and 5 squares, the ratio of triangles to squares is 5
5
and the ratio of squares to triangles is 6 .
Use the triangle at the right for Exercises 5–7.
5
13
5. Circle the ratio of the length of the longer leg to the length of the shorter leg.
5
13
5
12
12
13
13
12
12
5
13
5
6. Circle the ratio of the length of the shorter leg to the length of the hypotenuse.
5
13
5
12
12
13
13
12
12
5
13
5
7. Circle the ratio of the length of the longer leg to the length of the hypotenuse.
5
13
Chapter 8
5
12
12
13
13
12
12
5
210
13
5
12
Vocabulary Builder
Key Concept The Trigonometric Ratios
sine of /A 5
B
length of leg opposite/A
a
5
length of hypotenuse
c
b
cosine of /A 5
5
c
length of hypotenuse
length of leg opposite/A
tangent of /A 5
5
a
c
A
a
C
b
b
Draw a line from each trigonometric ratio in Column A to its corresponding ratio
in Column B.
Column A
Column B
a
c
b
a
b
c
8. sin B
9. cos B
10. tan B
11. Reasoning Suppose nABC is a right isosceles triangle. What would the tangent of
/B equal? Explain. Explanations may vary. Sample:
1. The legs would be congruent, so ba would equal 1.
_______________________________________________________________________
Problem 1 Writing Trigonometric Ratios
G
Got It? What are the sine, cosine, and tangent ratios for lG?
17
12. Circle the measure of the leg opposite /G.
8
15
17
T
15
8
R
13. Circle the measure of the hypotenuse.
8
15
17
14. Circle the measure of the leg adjacent to /G.
8
15
17
15. Write each trigonometric ratio.
sin G 5
15
opposite
5
hypotenuse
17
cos G 5
8
5
hypotenuse
17
tan G 5
15
opposite
5
8
211
Lesson 8-3
Problem 2 Using a Trigonometric Ratio to Find Distance
Got It? Find the value of w to the nearest tenth.
54
Below is one student’s solution.
17
w
w
17
cos 54î(17) â w
cos 54î â
9.992349289 Ƽ w
10 Ƽ w
16. Circle the trigonometric ratio that uses sides w and 17.
sin 548
cos 548
tan 548
17. What error did the student make?
w
Answers may vary. Sample: The student wrote cos 54° 5 17
rather
_______________________________________________________________________
w
than sin 54° 5 17
.
_______________________________________________________________________
18. Find the value of w correctly.
19. The value of w to the nearest tenth is 13.8 .
Problem 3 Using Inverses
Got It? Use the figure below. What is mlY to the nearest degree?
P
100
T
41
Y
20. Circle the lengths that you know.
hypotenuse
side opposite /Y
21. Cross out the ratios that you will NOT use to find m/Y .
sine
cosine
tangent
22. Underline the correct word to complete the statement.
If you know the sine, cosine, or tangent ratio of an angle, you can use the
inverse / ratio to find the measure of the angle.
Chapter 8
212
w
sin 54° 5 17
sin 54°(17) 5 w
13.7532889 N w
13.8 N w
23. Follow the steps to find m/Y .
Write the ratio.
1
tan Y â
100
41
Use the inverse.
2
3
Y â tanź1
( )
100
41
Use a calculator.
YƼ
67.70637084
24. To the nearest degree, m/Y < 68° .
Lesson Check • Do you UNDERSTAND?
Error Analysis A student states that sin A S sin X because the
lengths of the sides of kABC are greater than the lengths of the
sides of kXYZ. What is the student’s error? Explain.
Underline the correct word(s) to complete each sentence.
25. nABC and nXYZ are / are not similar.
B
Y
Z
35
X
C
35
A
26. /A and /X are / are not congruent, so sin 358 is / is not equal to sin 358.
27. What is the student’s error? Explain.
Answers may vary. Sample: The student did not look at the
_________________________________________________________________
measures of lA and lX. Congruent angles have equal sine ratios.
_________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
trigonometric ratios
sine
cosine
tangent
Rate how well you can use trigonometric ratios.
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213
Lesson 8-3
8-4
Angles of Elevation and
Depression
Vocabulary
Review
Underline the correct word(s) or number to complete each sentence.
1. The measure of a right angle is greater / less than the measure of an acute angle
and greater / less than the measure of an obtuse angle.
2. A right angle has a measure of 45 / 90 /180 .
3. Lines that intersect to form four right angles are parallel / perpendicular lines.
4. Circle the right angle(s) in the figure.
/ACB
/BAC
/CBA
/DBA
A
D
B
C
elevation (noun) el uh VAY shun
Related Word: depression
Definition: The elevation of an object is its height above a given level, such as eye
level or sea level.
Math Usage: Angles of elevation and depression are acute angles of right triangles
formed by a horizontal distance and a vertical height.
Complete each statement with the correct word from the list below. Use each word
only once.
elevate
elevated
elevation
5. John 9 his feet on a footstool.
elevated
6. The 9 of Mt McKinley is 20,320 ft.
elevation
7. You 9 an object by raising it to a higher position.
elevate
Chapter 8
214
Vocabulary Builder
Problem 1 Identifying Angles of Elevation and Depression
Got It? What is a description of l2 as it relates to the situation shown?
Write T for true or F for false.
T
8. /2 is above the horizontal line.
T
9. /2 is the angle of elevation from the person in the hot-air balloon to the bird.
F
10. /2 is the angle of depression from the person in the hot-air balloon to the bird.
F
11. /2 is the angle of elevation from the top of the mountain to the person in the
hot-air balloon.
12. Describe /2 as it relates to the situation shown.
Answers may vary. Sample: l2 is the angle of elevation from the
_______________________________________________________________________
person in the hot-air balloon to the bird.
_______________________________________________________________________
Problem 2 Using the Angle of Elevation
Got It? You sight a rock climber on a cliff at a 32° angle of elevation. Your
eye level is 6 ft above the ground and you are 1000 feet from the base of the
cliff. What is the approximate height of the rock climber from the ground?
13. Use the information in the problem to complete the problem-solving
model below.
Know
Angle of elevation
is
32
Need
Height of climber from
the ground
8.
Distance to the cliff
is
6
32
1000 ft
Eye level
Plan
Find the length of the
leg opposite 328 by
using tan 32 8 .
1000 ft.
Eye level is
Climber
6
ft.
ft
above the ground.
215
Lesson 8-4
14. Explain why you use tan 328 and not sin 328 or cos 328. Answers may vary. Sample:
The sine ratio involves two unknowns. The cosine ratio involves the
_______________________________________________________________________
hypotenuse and 1000, but I do not want to know the hypotenuse. The
_______________________________________________________________________
ratio that uses the unknown height and 1000 is the tangent ratio.
_______________________________________________________________________
15. The problem is solved below. Use one of the reasons from the list at
the right to justify each step.
d
tan 328 5 1000
Write the equation.
(tan 328) 1000 5 d
Solve for d.
Use a calculator.
Write the equation.
Solve for d.
d < 624.8693519
Use a calculator.
16. The height from your eye level to the climber is about 625 ft.
17. The height of the rock climber from the ground is about 631 ft.
Problem 3 Using the Angle of Depression
Got It? An airplane pilot sights a life raft at a 26° angle of depression. The airplane’s
altitude is 3 km. What is the airplane’s horizontal distance d from the raft?
26º
altitude 3 km
Not to scale
Angle of
elevation 26º
d
Angle of
depression
horizontal distance
Raft
19. Circle the equation you could use to find the horizontal distance d.
3
sin 268 5 d
3
cos 268 5 d
20. Solve your equation from Exercise 19.
tan 268 5 d3
d 5 tan3268
d 5 6.150911525
21. To the nearest tenth, the airplane’s horizontal distance from the raft is 6.2 km.
Chapter 8
216
3
tan 268 5 d
18. Label the diagram below.
Lesson Check • Do you UNDERSTAND?
Vocabulary How is an angle of elevation formed?
Underline the correct word(s) to complete each sentence.
22. The angle of elevation is formed above / below a horizontal line.
23. The angle of depression is formed above / below a horizontal line.
24. The measure of an angle of elevation is equal to / greater than / less than the
measure of the angle of depression.
Lesson Check • Do you UNDERSTAND?
Error Analysis A homework question says that the angle
of depression from the bottom of a house window to a
ball on the ground is 20°. At the right is your friend’s
sketch of the situation. Describe your friend’s error.
20˚
25. Is the angle that your friend identified as the angle of
depression formed by the horizontal and the line
of sight?
Yes / No
26. Is the correct angle of depression adjacent to or opposite
the angle identified by your friend?
Answers may vary. Sample: My friend identified the wrong angle. The
_______________________________________________________________________
correct angle of depression is below the horizontal line.
_______________________________________________________________________
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
angle of elevation
angle of depression
trigonometric ratios
Rate how well you can use angles of elevation and depression.
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217
Lesson 8-4
Law of Sines
8-5
Vocabulary
Review
1. Draw a line segment from each angle of the triangle to its opposite side.
C
b
a
A
B
c
2. Circle the correct word.
A ratio is the comparison of two quantities by
addition subtraction multiplication division sine (noun) syn
Related Words: triangle, side length, angle measure, opposite, cosine
Definition: In a right triangle, sine is the ratio of the side opposite a given acute angle
to the hypotenuse.
Example: If you know the measure of an acute angle of a right triangle and the
length of the opposite side, you can use the sine ratio to find the length of the
hypotenuse.
3. A triangle has a given acute angle. Circle its sine ratio.
hypotenuse
hypotenuse
opposite opposite
hypotenuse
opposite
4. A right triangle has one acute angle measuring 36.98. The length of the side adjacent
to this angle is 4 units, and the length of the side opposite this angle is 3 units. The
length of the hypotenuse is 5 units. Circle the sine ratio of the 36.98 angle.
43 Chapter 8
3
5
4
5
5
4
5
Vocabulary Builder
Law of Sines
For any nABC, let the lengths of the sides opposite angles A, B, and C be a, b, and c, respectively.
C
Then the Law of Sines relates the sine of each angle to the length of its opposite side.
sin B
sin C
sin A
a 5 b 5 c
b
5. If you know 2 angles and 1 side of a triangle, can you find all of the missing
measures? Explain.
a
A
B
c
Yes; since the sum of a triangle’s angles are 1808, you can find the third
angle; then use the Law of Sines to find the other 2 sides.
Problem 1 Using the Law of Sines (AAS)
Got It? In DABC, m/A 5 48, m/B 5 93, and AC 5 15.
What is AB to the nearest tenth?
C
398
6. Find and label m/C. 1808 5 488 1 938 1 39
c
7. Label side lengths a, b, and c. Which side is the length of AB? __________
a
8. Circle the equation which can be used to solve this problem. Explain
sin C
sin A
c 5 a sin C
sin B
c 5 b 93
B
sin B
sin A
5 a
b
15 b
48
A
c
Answers may vary. Sample: I want to pick the equation in which
______________________________________________________________
I have 3 out of the 4 values.
9. Replace the variables in the equation with values from DABC.
sin 398
c
5
sin 938
15
10. Find the sine values of the given angles, cross multiply, then solve for c.
c<
.6293 ) • ( 15 ) 9.453
( <
( .9986 )
9.5 units.
11. The length of AB is about Problem 2 Using the Law of Sines (SSA)
L
Got It? In DKLM , LM 5 9, KM 5 14, and m/L 5 105.
m
To the nearest tenth, what is m/K ?
12. Label the triangle with information from the problem
and the length of the sides as k, l, m.
219
K
105
14 5 l
95k
M
Lesson 8-5
13. Use the letter that represents the length of KM to write a pair of
ratios using some of the letters k, l, m, K, L and M.
sin L
l
5
sin K
k
14. Fill in the values in the equation from Exercise 13 and solve for sin K.
sin14105 5 sin9 K
sin K <
(.9659)(9)
14
< .6210
.6210
sin K < 15. Use your calculator and take the inverse sine of both sides of the equation to
find m/K .
sin
21
.6210 , therefore m/K < (sin K) < sin 21 38.4°
Problem 3 Using the Law of Sines to Solve a Problem
16. Underline the correct word to complete each sentence.
2nd
Base
In this problem, the solution is a side / angle .
60 ft
17. In order to use the Law of Sines what information
will you need that is missing and why?
I need to find the measure of the angle formed by second base,
______________________________________________________________
the right fielder, and first base because I need the measure of
______________________________________________________________
the angle opposite the given side to use the Law of Sines.
______________________________________________________________
18. Circle the equation you could use to solve for the missing solution.
sin 688
sin 728
60 5 a sin 728
sin 408
c 60 5
sin 728
sin 688
60 5 b 19. Fill in the blanks to complete the equation. Then solve the equation and find
the solution.
728
sin 60
0.9511
60
5
5
408
sin c
0.6428
c
c
(0.9511) < 60(0.6428)
40.6 feet.
Kimmy throws the ball about Chapter 8
72
68
To find the solution, I need to first find a missing side / angle .
Right-fielder
a
220 40.6
c < b
c
40
1st
Base
Got It? The right-fielder fields a softball between first base and second base as
shown in the figure. If the right-fielder throws the ball to second base, how far does
she throw the ball?
Lesson Check • Do you UNDERSTAND?
Reasoning If you know the three side lengths of a triangle, can you use the Law of Sines to
find the missing angle measures? Explain.
20. What do AAS, ASA, and SSA stand for? Match each term with its definition. Then tell
what the three terms have in common.
AAS
Side-Side-Angle
ASA
Angle-Angle-Side
SSA
Angle-Side-Angle
They all include sides and angles.
______________________________________________________________
21. If you know only the three side lengths of a triangle, can you use the Law of Sines to find
the missing angle measures? Explain.
Answers may vary. Sample: No; To use the Law of Sines, you
______________________________________________________________
need 2 angles and 1 side or 2 sides and 1 angle.
______________________________________________________________
Error Analysis In DPQR, PQ 5 4 cm, QR 5 3 cm, and m/R 5 75.
Your friend uses the Law of Sines to write
Explain the error.
sin 758
sin x8
5 4 to find m/Q.
3
P
22. Label the diagram with the given information. Did your friend correctly match the
angles and the sides?
4
Answers may vary. Sample: The side length opposite /R is 4,
________________________________________________________________________
not 3. /Q is not opposide side length 4.
________________________________________________________________________
75
R
x
3
Q
Math Success
Check off the vocabulary words that you understand.
Law of Sines
ratio
inverse sine
Rate how well you can use the Law of Sines.
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6
8
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221
Lesson 8-5
Law of Cosines
8-6
Vocabulary
Review
Look at DABC.
C
b
a
A
B
c
b and c
1. Name the sides that are adjacent to angle A. ___________
b
2. Which side is opposite of angle B? ______
3. Identify each angle measure as acute, right, or obtuse.
458
acute
________
1008
obtuse
________
908
right
________
Cosine (noun) KOH syn
Related Word: triangle, side length, angle measure, opposite, sine
Definition: In a right triangle, cosine is the ratio of the side adjacent to a given acute
angle to the hypotenuse.
Example: If you know the measure of an acute angle of a right triangle and the
length of the adjacent side, you can use the cosine ratio to find the length of the
hypotenuse.
4. A triangle has a given acute angle. Circle its cosine ratio.
hypotenuse
hypotenuse
hypotenuse
opposite 5. A right triangle has one acute angle measuring 53.18, the length of the side adjacent
to this angle is 9 units, and the length of the side opposite this angle is 12 units. The
length of the hypotenuse is 15 units. Circle the cosine ratio of the 53.18 angle.
9
12 Chapter 8
12
15 9
15 15
9
15
Vocabulary Builder
Law of Cosines
For any nABC with side lengths a, b, and c opposite angles A, B, and C, respectively, the
Law of Cosines relates the measures of the triangles according to the following equations.
a2 5 b2 1 c2 2 2bc cos A
b2 5 a2 1 c2 2 2ac cos B
c2 5 a2 1 b2 2 2ab cos C
C
b
a
A
d 2 5 f 2 1 e 2 2 2de cos D
f 2 5 d 2 1 e 2 2 2de cos F
e 2 5 d 2 1 f 2 2 df cos E
B
c
6. Circle the equation that is true for DDEF .
E
d
f
F
D
e
Problem 1 Using the Law of Cosines (SAS)
l
M
Got It? In DLMN, m/L 5 1048, LM 5 48, and LN 5 29. Find MN to
N
the nearest tenth.
48
n
7. Label the sides of DLMN with the letters l, m, and n.
104
29 m
L
8. Use the information in the problem to complete the problem-solving model below.
Know
Need
LM is opposite / N
MN 5 letter
I
n
LM 5 48 5 letter
LN is opposite / M
LN 5 29 5 letter
Plan
m
An equation using
letters l, m, n, and
/L.
Because you know
m/ L and need
MN, substitute the angle
measure and the two
side lengths into the
equation and solve for l.
9. Find MN by solving for l.
a. I 2 5 m 2 1 n 2 2 2( m )( n ) cos /L
a. Write an equation using l, m, n, and /L.
b. l 2 5 29 2 1 48 2 2 2( 29 )( 48 ) cos 104 b. Substitute the values from the triangle.
< 841 1 2304 2 (2784 ? 20.242)
c. l2 c. Use the Order of Operations and
solve for l 2 .
< 3145 1 673.728
l2 l2 < 3818.728 d. l < 61.8 5 MN
d. Take the square root of both sides.
223
Lesson 8-6
Problem 2 Using the Law of Cosines (SSS)
T
u
Got It? In DTUV above, find m/T to the nearest tenth of a degree.
10. Label the sides of the triangle with t, u, and r.
U
11. Solve for m/T following the given STEPS.
6.7
4.4
v
t
7.1
V
t 2 5 u 2 1 v 2 2 2( u )( v ) cos T Write an equation using the
Law of Cosines.
7.1 2 5 6.7 2 1 4.4 2 2 2( 6.7 )( 4.4 ) cos T Substitute the values from
the triangle.
50.41 5 44.89
1 19.36
Simplify by squaring and
multiplying.
2 58.96 cos T 50.41
50.41 2
64.25 5 2 58.96 cos T Get coefficient of cos T and
cos T alone.
213.84
58.96
cos 21 0.234735
5 cos T
Divide by the coefficient
of cos T.
< T
Take the inverse cosine of
both sides of the equation.
m/T < 76.4 Problem 3 Using the Law of Cosines to Solve a Problem
Got It? 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 118W (118west of south) from the
campsite. To the nearest tenth of a mile, how far apart are the two groups?
12. Label the model with information from the problem and
letter the angles and sides.
West
campsite
c
b = 1.9
miles
a = 1.4 miles
11
South
13. Find the measure of the angle that is the complement of
the 118angle.
798
908 2 118 5 Chapter 8
5 64.25
2 58.96 cos T 14. Write and solve an equation for finding the distance between the two groups.
c2 5 1.42 1 1.92 2 2(1.4)(1.9) cos(79)
c2 5 1.96 1 3.61 2 5.32 (.191)
c2 5 5.57 2 1.02
c2 5 4.55
c < 2.13 miles
Lesson Check • Do you UNDERSTAND?
Writing Explain how you choose between the Law of Sines and the Law of Cosines
when finding the measure of a missing angle or side.
15. Write C if you would use the Law of Cosines to find a missing measure in a triangle
or S if you would use the Law of Sines.
C
The lengths of two sides and the measure of the included angle are given.
Find the length of the third side.
C
The lengths of three sides are given. Find the measure of one angle.
S
The measures of two angles and the length of the included side are given.
Find the length of another side.
16. Explain how to choose between the Law of Sines and the Law of Cosines in solving
a triangle.
Answers may vary. Sample: Use the Law of Cosines when you know SAS
________________________________________________________________________
or SSS. Use the Law of Sines when you know SSA or ASA.
________________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
Law of Cosines
Law of Sines
trigonometry
Rate how well you can use the Law of Cosines.
Need to
review
0
2
4
6
8
Now I
get it!
10
225
Lesson 8-6
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