GUIDED PRACTICE
Vocabulary Check
Concept Check
Skill Check
✓
✓
✓
3 APPLY
In Exercises 1–3, use the diagram at the right.
? of ML and JL.
1. In the diagram, KL is the geometric mean
2. Complete the following statement:
? ~¤ ? . KML, JMK
¤JKL ~ ¤
M
L
ASSIGNMENT GUIDE
BASIC
Day 1: pp. 531–534 Exs. 11–30,
33, 34, 41, 42, 44–50 even
K
J
3. Which segment’s length is the geometric mean
Æ
of ML and MJ? MK
AVERAGE
Day 1: pp. 531–534 Exs. 11–30,
33–36, 41, 42, 44–50 even
In Exercises 4–9, use the diagram above. Complete the proportion.
ADVANCED
?
KM
4. = JM
JK
KL
JM
JK
5. = JK
?
JL
?
LK
6. = LK
LM
JM
KM
7. = ?
LM
LK
JK
8. = KM
LM
?
?
MK
9. = LK
JK
MJ
KM
10. Use the diagram at the right. Find DC.
Day 1: pp. 531–534 Exs. 11–30,
32–36, 41–43, 44–50 even
JL
BLOCK SCHEDULE
(WITH CH. 8 ASSESS.)
pp. 531–534 Exs. 11–30, 33–36,
41, 42, 44–50 even
F
Then find DF. Round decimals to the
nearest tenth. 53.4; 48.3
72
65
D
97
E
C
EXERCISE LEVELS
Level A: Easier
11–15
Level B: More Difficult
16–33, 37–42
Level C: Most Difficult
34–36, 43
PRACTICE AND APPLICATIONS
q
STUDENT HELP
SIMILAR TRIANGLES Use the diagram.
Extra Practice
to help you master
skills is on p. 819.
11. Sketch the three similar triangles in the
diagram. Label the vertices. See margin.
T
12. Write similarity statements for the
three triangles. ¤QRS ~ ¤QST ~ ¤SRT
R
HOMEWORK CHECK
To quickly check student understanding of key concepts, go over
the following exercises: Exs. 12,
14, 18, 20, 22, 28, 30, 34. See also
the Daily Homework Quiz:
• Blackline Master (Chapter 9
Resource Book, p. 24)
•
Transparency (p. 64)
S
USING PROPORTIONS Complete and solve the proportion.
1
x
?
4
x
5
x
13. = 20; 33 }3}
14. = 9; 6
15. = 20
12
x
?
x
?
3; 兹15
苶
x
20
16. ¤XYZ ~ ¤XZW ~
¤ZYW ; ZW
x
3
x
12
4
9
17. ¤QRS ~ ¤QST ~
¤SRT; RQ
18. ¤EFG ~ ¤EGH ~
¤GFH ; EH
STUDENT HELP
11.
2
COMPLETING PROPORTIONS Write similarity statements for the three
similar triangles in the diagram. Then complete the proportion.
QT
SQ
17. = SQ
?
?
XW
16. = YW
ZW
Y
HOMEWORK HELP
Example 1: Exs. 11–31
Example 2: Exs. 11–31
Example 3: Ex. 32
X
S
Q
R
T
S
S
T
R
?
EG
18. = EG
EF
q
W
Q
E
H
F
T
Z
S
R
G
9.1 Similar Right Triangles
531
531
MATHEMATICAL REASONING
EXERCISE 19 The altitude to the
hypotenuse of a right triangle is the
geometric mean between each
segment of the hypotenuse. This
property can be used without first
identifying the similar triangles. Ask
students to solve the proportion
16 12
ᎏᎏ = ᎏᎏ to find DB in this triangle.
12 DB
STUDENT HELP NOTES
FINDING LENGTHS Write similarity statements for three triangles in the
diagram. Then find the given length. Round decimals to the nearest tenth.
21. ¤JKL ~ ¤JLM ~
• Drag point C so that måC ≠ 90°.
AB BC AB
Calculate the ratios }}, }}, }},
BC BD AC
AC
AB BC
and }}. Notice that }} ≠ }}
AD
BC B D
AB AC
}
}
}
}
and
≠ .
AC A D
• Conclude that Theorem 9.3 is
true only for a right triangle.
M
16
A
15 K
20
D
E
B
H
32
F
25
J
L
¤EFG ~ ¤EGH ~ ¤GFH; 16
22. Find QS.
24. ¤EGH ~ ¤EHF ~ ¤HGF;
3兹14
苶 ≈ 11.2
q
23. Find CD.
24. Find FH.
C
R
40
F
25
32
7
G
T
S
¤QRS ~ ¤QTR ~ ¤RTS; 50
A
4
D
4
B
¤ABC ~ ¤ACD ~ ¤CBD; 4
E
H
xy USING ALGEBRA Find the value of each variable.
25.
26.
3
12
3兹3苶
INT
STUDENT HELP
NE
ER T
20
7
4
7
10}}
28.
30.
29.
14
z
c
y
e
32
d
x
24
1
2
x = 42 }}, y = 40, z = 53 }}
3
3
16
31. about 76 cm; ¤ABC and
苶
c = 12.25, d = 3.75, e = 7兹15
}
¤ADC are congruent
4
right triangles by the SSS 31.
KITE DESIGN You are designing a diamondA
Congruence
Post., so
Æ
shaped
kite.
You
know
that
AD
=
44.8
centimeters,
AC is a perpendicular
Æ
DC = 72 centimeters, and AC = 84.8 centimeters.
bisector of BD . By
Æ
You want to use a straight crossbar BD. About how
B
Geometric Mean
long should it be? Explain.
Theorem 9.3, the altitude
from D to hypotenuse
Æ
Æ
AC divides AC into
32.
ROCK CLIMBING You and a friend
Not drawn to scale
segments of lengths
want to know how much rope you need
23.67 cm and 61.13 cm.
By Geometric Mean
to climb a large rock. To estimate the
Theorem 9.2, the length
height of the rock, you use the method
of the altitude to the
from Example 3 on page 530. As shown
hypotenuse of each right
at the right, your friend uses a square to
triangle is about 38 cmÆ
line up the top and the bottom of the rock.
long, so the crossbar BD
You measure the vertical distance from
should be about 2 • 38, or
76 cm long.
the ground to your friend’s eye and the
about 64.4 ft
532
m
9.6
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with Exs. 28–30.
5
x
9
x
27.
16
distance from your friend to the rock.
Estimate the height of the rock.
532
21. Find JK.
G
¤ABC ~ ¤ACD ~ ¤CBD; 9
TEACHING TIPS
EXERCISE 31 Remind students
that they can prove that the diagonals of a kite are perpendicular and
that A
苶C苶 bisects B
苶D
苶.
AD
B C BD AC
AB BC
Notice that }} = }} and
BC BD
AB AC
}} = }}.
AC A D
20. Find HF.
C
12
Homework Help Students
can find help for Exs. 28–30 at
www.mcdougallittell.com.
The information can be printed
out for students who don’t have
access to the Internet.
40. A good answer should include
these steps:
• Use the right triangle ABC from
Ex. 38. Measure AD, BD, AB, BC,
and AC.
• Use the calculate feature to
calculate the values of the
AC
AB BC AB
ratios }}, }}, }}, and }},
19. Find DB.
1024
15
¤LKM; }} ≈ 68.3
Chapter 9 Right Triangles and Trigonometry
xⴙ9
8
3
18
D
C
h
18 ft
1
5 2 ft
33. FINDING AREA Write similarity statements
STUDENT HELP
B
for the three similar right triangles in the
diagram. Then find the area of each triangle.
Explain how you got your answers.
Look Back
For help with finding
the area of a triangle,
see p. 51.
1.6 and DC = 1.2, so the
A
2m
2
(1.2) = 0.96 m , and the
area of ¤CBD =
1.5 º 0.96 = 0.54 m 2.
C
PROVING THEOREMS 9.1, 9.2, AND 9.3 In Exercises 34–36, use the
diagram at the right.
1
area of ¤ACD = }} (1.6) x 34. Use the diagram to prove Theorem 9.1 on page 527.
2
Look Back As students look back
to p. 51, remind them that either
leg of a right triangle can be used
as the altitude in the area formula.
1.5 m
33. ¤ABC ~ ¤ACD ~
¤CBD; area of ¤ABC =
1
}} (2)(1.5) = 1.5 m 2; AD =
2
STUDENT HELP NOTES
2.5 m D
C
(Hint: Look back at the plan for proof on page 528.)
GIVEN 䉴 ¤ABC is a right triangle;
Æ
Æ
altitude CD is drawn to hypotenuse AB.
A
D
B
34. Proof that ¤CBD ~
PROVE 䉴 ¤CBD ~ ¤ABC, ¤ACD ~ ¤ABC,
¤ABC: All rt. √ are £ ,
and ¤CBD ~ ¤ACD.
so ™BDC £ ™BCA.
™B £ ™B, so ¤CBD ~
35. Use the diagram to prove Theorem 9.2 on page 529. From Ex. 34, ¤CBD ~
¤ABC by the AA
¤ACD. Corresponding
Similarity Post. Proof that
GIVEN 䉴 ¤ABC is a right triangle;
side lengths are in
Æ
Æ
¤ACD ~ ¤ABC : All rt. √
altitude CD is drawn to hypotenuse AB.
CD
BD
are £ , so ™CDA £
proportion, so }} = }} .
AD
CD
™BCA. ™A £ ™A, so
BD
CD
PROVE 䉴 = ¤ACD ~ ¤ABC by the
CD
AD
AA Similarity Post. Proof
36. Use the diagram to prove Theorem 9.3 on page 529.
that ¤CBD ~ ¤ACD:
Æ
™B is complementary to
GIVEN 䉴 ¤ABC is a right triangle; altitude CD is
™DCB because the acute
Æ
drawn to hypotenuse AB.
√ of a right ¤ are
complementary. In ¤ACB,
AB
BC
AB
AC
PROVE 䉴 = and = ™ACD is complementary
BC
BD
AC
AD
to ™DCB because two
adjacent √ that form a rt.
USING TECHNOLOGY In Exercises 37–40, use geometry software.
™ are complementary.
You will demonstrate that Theorem 9.2 is true only for a right triangle.
So, ™B £ ™ACD
Follow the steps below to construct a triangle.
because two √
complementary to the
1
Draw a triangle and label its vertices
same ™ are £. All rt. √
A, B, and C. The triangle should not be
C
are £ , so ™BDC £
a
right triangle.
™CDA. Thus, ¤CBD ~
Æ
Æ
¤ACD by the AA
2
Draw altitude CD from point C to side AB.
Similarity Post.
36. From Ex. 34, ¤CBD ~
¤ABC. Corresponding
side lengths are in
proportion, so
BC
AB
}} = }} . Also, from
BD
BC
Ex. 34, ¤ACD ~ ¤ABC.
Corresponding side
lengths are in
AC
AB
proportion, so }} = }}.
AC
AD
39. The ratios are equal
when the triangle is a
right triangle, but are not
equal when the triangle
is not a right triangle.
3
Æ Æ
Æ
Measure ™C. Then measure AD, CD, and BD.
A
D
TEACHING TIPS
EXERCISE 38 Dragging point C
until the angle measures 90° uses
the dynamic feature of the geometry drawing software. It is important
that students first calculate the
ratios in Ex. 37 and leave them
visible on the screen so that they
can see the changes as ⬔C
changes.
ENGLISH LEARNERS
EXERCISES 37–40
Understanding English language
directions for programming a
graphing calculator can be difficult
for English learners. You may want
to have English learners work with
an English fluent partner to follow
the steps for constructing a triangle
and solving Exercises 37–40.
B
BD
CD
37. Calculate the values of the ratios and .
CD
AD
What does Theorem 9.2 say about the values of
these ratios? The values of the ratios will vary, but will not be equal. The theorem
says that these ratios are equal.
38. Drag point C until m™C = 90°. What happens to
BD
CD
the values of the ratios and ? The ratios are equal.
CD
AD
ADDITIONAL PRACTICE
AND RETEACHING
39. Explain how your answers to Exercises 37 and 38 support the conclusion that
Theorem 9.2 is true only for a right triangle.
40. Use the triangle you constructed to show that Theorem 9.3 is true only for a
right triangle. Describe your procedure. See margin.
9.1 Similar Right Triangles
533
For Lesson 9.1:
• Practice Levels A, B, and C
(Chapter 9 Resource Book, p. 14)
• Reteaching with Practice
(Chapter 9 Resource Book, p. 17)
•
See Lesson 9.1 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
533
Test
Preparation
4 ASSESS
41. MULTIPLE CHOICE Use the diagram at the
right. Decide which proportions are true. D
DAILY HOMEWORK QUIZ
Transparency Available
Use the diagram.
DB
DA
I. ᎏᎏ = ᎏᎏ
DC
DB
BA
CB
II. ᎏᎏ = ᎏᎏ
CB
BD
BA
CA
III. ᎏᎏ = ᎏᎏ
CA
BA
DB
DA
IV. ᎏᎏ = ᎏᎏ
BC
BA
A
¡
A
B
¡
I only
C
D
A
C
¡
II only
B
D
¡
I and II only
I and IV only
42. MULTIPLE CHOICE In the diagram above, AC = 24 and BC = 12.
Find AD. If necessary, round to the nearest hundredth. C
A
¡
B
★ Challenge
D
43.
C
1. Write similarity statements for
the three triangles.
†ADC Í †ABD Í †DBC
B
¡
6
C
¡
16.97
Two methods for indirectly measuring the height of a building are
shown below. For each method, describe what distances need to be measured
directly. Explain how to find the height of the building using these
measurements. Describe one advantage and one disadvantage of each
method. Copy and label the diagrams as part of your explanations.
Use the method
described in Example 3 on
page 530.
Use the method
described in Exercises 55 and 56
on page 486.
Method 1
AD CB
ᎏᎏ = ᎏᎏ BD
DC
?
Find the value of the variable.
3.
Method 2
B
R
Not drawn to scale
8
x
Not drawn to scale
15
64
}} ≈ 4.3
15
N
A
EXTRA CHALLENGE
Challenge problems for
Lesson 9.1 are available in
blackline format in the Chapter 9
Resource Book, p. 21 and at
www.mcdougallittell.com.
ADDITIONAL TEST
PREPARATION
1. OPEN ENDED Draw and label
a right triangle and the altitude
to the hypotenuse. Identify three
similar triangles in the diagram.
Sample answer:
A
C
M P
D
EXTRA CHALLENGE NOTE
47. If the measure of one of
xy SOLVING EQUATIONS Solve the equation. (Skills Review, p. 800, for 9.2)
the angles of a triangle is
45. 14 + x 2 = 78 8, º8
46. d 2 + 18 = 99 9, º9
greater that 90°, then the 44. n2 = 169 13, º13
triangle is obtuse; true.
LOGICAL REASONING Write the converse of the statement. Decide
48. If the corresponding
whether the converse is true or false. (Review 2.1)
angles of two triangles
47. If a triangle is obtuse, then one of its angles is greater than 90°.
are congruent, then the
triangles are congruent; 48. If two triangles are congruent, then their corresponding angles are congruent.
false.
FINDING AREA Find the area of the figure. (Review 1.7, 6.7 for 9.2)
36 in.2
6 in.
12 in.
B
534
51.
50.
†ACD Í †CBD Í †ABC
534
S
MIXED REVIEW
D
43. See Additional Answers beginning on page AA1.
q
www.mcdougallittell.com
49.
C
20.78
Writing
See margin.
2. Complete the proportion.
D
¡
18
Chapter 9 Right Triangles and Trigonometry
4.5 cm
7 cm
31.5 cm 2
12 m
5m
62.5 m 2
13 m