Practice
b.
ACTIVITY 3.1
1. Explain why the following triangles are similar
to one another.
y
4n
x
3n
4
z
x
3
y
The arrows indicate
parallel lines.
7
15 + x
20 m
40°
32 m
Y
30 m
Q
2. x = 25m, y = 15m, z = 58°
3. Answers may vary. Check
students’ rectangles.
The sides should be in the
ratio 7:4.
ACTIVITY 3.3
___
___
___
6. Given the diagram with LD AE NT and
segment measures as shown, determine the
following measures. Show your work.
R
X
1. Corresponding sides are in
proportion, and corresponding
angles are congruent.
5m
c.
2. The following triangles are similar. Determine
the values of x, y, and z.
Activity 3.2
y
z - 18°
24 m
4. x ≈ 2.042 ft
60 cm
48 cm
S
S
D
E
T
5a. SSS
b. SAS
45 cm
L
3. Sketch a figure that is similar but not congruent
to a rectangle with length 14 in. and width 8 in.
ACTIVITY 3.2
a. SL
4. Solve for x in the following figure.
22 ft.
8 cm
A
b. LD
c. AA
16 cm
N
c. ET
Activity 3.3
48
x
6a. ___
= _____
60
x+8
x = 32; SL = 32 cm
d. NT
7. Given CEL with measures as shown,
determine x. Show your work.
4x + 1
48
x
b. ___
= ___
45
60
x = 36; LD = 36 cm
48 cm
© 2010 College Board. All rights reserved.
18 ft.
7.5 ft.
U
C
5. State the theorem or postulate that allows you
to determine similarity in the following pairs of
triangles.
36 cm
E
60
40
___
c. ___
x = 16
x = 24; ET = 24 cm
42 cm
x
45
60
___
d. ___
x = 84
x = 63; NT = 63 cm
L
a.
12 cm
24 cm
ACTIVITY 3.4
12 cm
8 cm
36 cm
4 cm
8. The ratio of similarity of corresponding sides of
2 . If the coordinates of
STU and VWX is __
3
S, T, and V are S (-4, 6), T(2, 6), and V(5, 7),
determine a possible pair of coordinates for W.
HIJ = H(8, -2), I(12, -2), J(-9, 8)
LMN = L(-8, -8), M(-6, -8), N(-8, -1)
Unit 3 • Similarity, Right Triangles, and Trigonometry
x = ___
12
7. ___
42
36
x = 14
Activity 3.4
8. Possible coordinates: (14, 7)
or (-4, 7)
9. Determine if HIJ ˜ LMN.
9. Corresponding side lengths
are not in proportion so the
triangles are not similar.
269
1/21/10 2:11:08 PM
© 2010 College Board. All rights reserved.
269-271_SB_Geom_3-PRAC_SE.indd 269
UNIT 3 PRACTICE
Activity 3.1
14 m
z
7n
W
UNIT 3
Unit 3 • Similarity, Right Triangles, and Trigonometry
269
UNIT 3
Practice
Activity 3.5
ACTIVITY 3.5
10a. PM = 6 in.; x2 + 82 = 102;
x=6
10. Given RAM as shown.
b. RP = 10_23_ in.; _68_ = _8x_;
x = 10_2_
3
c. RA = 13_13_ in.;
32 2
40
___
+ 64 = x2; ___
=x
3
3
( )
Activity 3.6
11.
__
5√2
P
a. Determine PM.
b. Determine RP.
c. Determine RA.
8 in
R
M
10
A
ACTIVITY 3.6
11. Find the length of the hypotenuse of an isosceles
right triangle with leg length 5 centimeters. Give
the exact answer.
18. The perimeter of a square is 40 cm. Find the
length of a diagonal.
19. Find the perimeter of a square, to the nearest
tenth, if the length of its diagonal is 14 inches.
20. Find d and e.
60°
cm
13. 30 inches
14a. right
b. acute
c. not a triangle
d. obtuse
__
__
15. AB = √2 ; BC =7√2 ;
AC = 10; AB + BC >10
so a triangle exists.
AC 2 = AB 2 + BC 2 so the
triangle is a right triangle.
Activity 3.7
16. a = 3; b =
__
e
8
12. Find the length of the altitude drawn from the
vertex of an isosceles triangle with side lengths
13 in., 13 in., and 24 in.
12. 5 inches
e. right
17. The measure of each leg of an isosceles right
triangle is 5. Find the measure of the hypotenuse.
d
13. An isosceles trapezoid has bases that are 7 inches
and 13 inches long. The height of the trapezoid is
4 inches. Find the perimeter of the trapezoid.
21. The longer leg of a 30°– 60°– 90° triangle is
6 inches. What is the length of the hypotenuse?
14. Tell whether a triangle can be formed having
the following side lengths. If a triangle can be
formed, tell whether it is right, acute, or obtuse.
22. The length
of an altitude of an equilateral triangle
__
is 2 √3 inches. Find the length of a side of the
triangle.
a.
b.
c.
d.
e.
9, 40, 41
4, 5, 6
5, 12, 18
9, 9, 13
27, 36, 45
23. One side of an equilateral triangle is 8 cm. Find
the length of the altitude.
24. The perimeter of an equilateral triangle is
36 inches. Find the length of an altitude.
15. Use the given vertices to determine whether
ABC is a right triangle. Explain your reasoning
and show the calculations that led to your
answer. A(2, 7) B(3, 6) C(–4, –1)
25. Find the perimeter of the trapezoid.
© 2010 College Board. All rights reserved.
UNIT 3 PRACTICE Continued
18 cm
6 cm
60°
__
3√2
ACTIVITY 3.7
26. Find the perimeter of PQR.
16. Find a and b.
17. 5√2
R
__
18. 10√2 cm
19. 39.6 in.
__
60°
b
a
10 √ 3
20. d = 8√3 ; e = 16
__
45°
12
__ inches, or 4√3 inches
21. ___
√3
P
30°
Q
3
22. 4 inches
__
23. 4√3 cm
__
24. 6√3 in.
25. 54 cm
270 SpringBoard® Mathematics with Meaning™ Geometry
__
__
27. a = 3√6 ; b =
269-271_SB_Geom_3-PRAC_SE.indd 270
__
6√2
270 SpringBoard® Mathematics with Meaning™ Geometry
1/21/10 2:11:11 P2
© 2010 College Board. All rights reserved.
26. 60 + 20√3
Practice
b
6
a
12
ACTIVITY 3.8
28. Draw RST with right angle S. Identify the
hypotenuse, adjacent leg, and opposite leg for ∠T.
c. 0.306
15
30a. ___
17
15
b. ___
8
Q
a. cos 49° b. sin 75° c. tan 17°
30. Using ABC below, write a ratio for the following.
R
17
c. ___
8
10
y
17
b. 0.966
N
x
35. Find x and y. Round to the nearest tenth.
29. Use your calculator to evaluate the following.
Round to 3 decimal places.
A
29a. 0.656
y
36°
M
60°
31. D__
P
x
√3
32a. ___
2
36. Find each of the following using the given
triangle. Round to the nearest tenth.
B
15
31. In the diagram below, which trigonometric ratio
4?
corresponds to __
5
E
© 2010 College Board. All rights reserved.
5
D
a. cos E
b. sin D
F
c. tan D
d. cos D
32. Label the 30°– 60°– 90° triangle shown below
with hypotenuse length 8 cm and label side
lengths. Use your triangle to find the exact value
of each ratio. Simplify all radicals.
a.
b.
c.
d.
e.
f.
cos 30°
sin 30°
tan 30°
sin 60°
cos 60°
tan 60°
70°
c. AC
B
C
7.2
38. A ramp at the loading dock of an automobile
manufacturing plant makes a 29° angle with
the ground. The bottom end of the ramp is
30 meters from the building.
a. Draw and label a diagram to illustrate the
situation.
b. Set up and solve an equation to find the
length of the ramp. Round to tenths.
30°
__
√3
c. ___
3
√3
d. ___
2
e. _12_
f. √3
__
33. cos T = _35_; tan T = _43_
b. AB
37. A kite string is 200 meters long. Find the height
of the kite if the string makes an angle of 38°
with the ground.
3
4
a. m∠C
b. _12_
__
A
C
___
28. hypotenuse:
RT ; opposite
___
___
leg: RS ; adjacent leg: ST
O
30°
45°
UNIT 3 PRACTICE Continued
Activity 3.8
34. Find x and y. Round to the nearest tenth.
27. Find a and b.
a. cos B.
b. tan A
c. csc B
UNIT 3
39. In an isosceles triangle, the base is 32 cm long
and the base angles are 56°. How long are the
legs?
34. x = 9.7; y = 7.1
35. x = 11.5; y = 5.8
36a. 20°
b. AB = 2.6
c. AC = 7.7
37. 123.1 meters
38a.
r
29°
30 m
b. r = 34.3 meters
39. 28.6 cm
4 ; find
33. In RST with right angle R, if sin T = __
5
cos T and tan T.
Unit 3 • Similarity, Right Triangles, and Trigonometry
1/21/10 2:11:13 PM
© 2010 College Board. All rights reserved.
269-271_SB_Geom_3-PRAC_SE.indd 271
PM
271
Unit 3 • Similarity, Right Triangles, and Trigonometry
271