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Chapter
7
Chapter Review
7
Chapter Review
Vocabulary Review
Resources
Cross-Product Property (p. 367)
extended proportion (p. 367)
geometric mean (p. 392)
golden ratio (p. 375)
golden rectangle (p. 375)
indirect measurement (p. 384)
proportion (p. 367)
ratio (p. 366)
scale (p. 368)
scale drawing (p. 368)
similar (p. 373)
similarity ratio (p. 373)
Student Edition
Extra Skills, Word Problems, Proof
Practice, Ch. 7, p. 728
English/Spanish Glossary, p. 779
Postulates and Theorems, p. 770
Table of Symbols, p. 763
Choose the correct term to complete each sentence.
1. Two polygons are 9 if corresponding angles are congruent and corresponding
sides are proportional. similar
2. The 9 states that the product of the extremes is equal to the product of the
means. Cross-Product Property
Vocabulary and Study Skills
worksheet 7F
Spanish Vocabulary and Study
Skills worksheet 7F
Interactive Textbook Audio
Glossary
Online Vocabulary Quiz
3. A 9 is a rectangle that can be divided into a square and a rectangle that is
similar to the original rectangle. golden rectangle
similarity ratio
4. The ratio of the lengths of corresponding sides of two similar figures is the 9.
PHSchool.com
For: Vocabulary quiz
Web Code: auj-0751
5. A 9 is a statement that two ratios are equal. proportion
6. Finding distances using similar triangles is called 9. indirect measurement
7. The length and width of a golden rectangle are in the 9. golden ratio
Skills and Concepts
To write ratios and solve
proportions
A ratio is a comparison of two quantities by division. You can write the ratio of
a to b or a ; b as the quotient ba when b 2 0.
A proportion is a statement that two ratios are equal. According to the Properties
of Proportions, ba 5 dc is equivalent to
(1) ad = bc
(2) b 5 d
c
a
(3) a 5 b
c
d
b
c1d
(4) a 1
b 5 d
Property 1, above, illustrates the Cross-Product Property, which states that the
product of the extremes is equal to the product of the means.
Spanish Vocabulary/Study Skills
Vocabulary/Study Skills
Name
When three or more ratios are equal, you can write an extended proportion.
L3
Date
For use with Chapter Review
Study Skill: Always read direction lines before doing any exercises. What
you think you are supposed to do with an activity may be quite different
than what the directions call for.
In a scale drawing, the scale compares each length in the drawing to the actual
length being represented.
Circle the word that best completes the sentence.
1. A number in (standard form, scientific notation) is written as a product
of two factors in the form a 3 10n, where n is an integer and 1 # a , 10.
Dollhouses Dollhouse furnishings come in different sizes depending on the size of
the dollhouse. For each exercise, write a ratio of the size of the dollhouse item to
the size of the larger item.
8. dollhouse sofa: 112 in. long;
real sofa: 6 ft long 1 : 48
Class
8D: Vocabulary
ELL
2. Each number in a sequence is called a (term, constant).
3. In a(n) (arithmetic, geometric) sequence you multiply a term in the sequence
by a fixed number.
4. The (Substitution, Elimination) method is a way of solving systems of
equations by replacing one variable with an equivalent expression.
5. A system of linear equations has (no solution, many solutions) when the
graphs of the equations are parallel lines.
6. In the function f(x) = 5x, as the values of the domain increase, the values
of the range (increase, decrease).
9. dollhouse piano: 134 in. high
real piano: 3 ft 6 in. high 1 : 24
7. When a bank pays interest on both the principal and interest the account
has already earned, the bank is paying (simple, compound) interest.
8. A(n) (interest, growth) period is the length of time over which interest
is calculated.
© Pearson Education, Inc. All rights reserved.
7-1 Objectives
9. In a relation the first set of coordinates in the ordered pairs is called
the (domain, range).
p
If q 5 25, tell whether each equation must be true. Explain. 10–13. See margin.
q
p
q
10. 2q = 5p
11. 25 5 p
12. 5q= 2p
13. 2 5 5
10. A base and an exponent are the two parts of a (symbol, power).
11. Lines in the same plane that intersect to form a 90° angle are said to
be (perpendicular, parallel).
12. The (median, mode) of a collection of data is the data item that occurs
most often.
13. The result of a single trial is called the (outcome, probability).
14. Each item in a matrix is called a(n) (term, element).
15. In the exponential function y = a ? bx, a ⬎ 0 and b ⬎ 1, the base (b)
is the (decay, growth) factor.
Chapter 7 Chapter Review
407
16. –2, 4, 12, 34, –8, 6 are examples of (real numbers, integers).
32
10. True; use the CrossProduct Prop.
11. True; the cross product
eq. is equivalent to the
original proportion.
Reading and Math Literacy Masters
Algebra 1
12. False; the cross product
eq. is not equivalent to
the original proportion.
13. True; the cross product
eq. is equivalent to the
original proportion.
407
7-2 and 7-3 Objectives
To identify and apply
similar polygons
To use AA, SAS, and SSS
similarity statements
To apply AA, SAS, and
SSS similarity statements
Similar polygons have congruent corresponding angles and proportional
corresponding sides. The ratio of the lengths of corresponding sides is the
similarity ratio.
A golden rectangle is a rectangle that can be divided into a square and
a rectangle that is similar to the original rectangle. In any golden rectangle,
the length and the width are in the golden ratio, which is
or about 1.618 ; 1.
1 1 #5
: 1,
2
If two angles of one triangle are congruent to two angles of another triangle,
then the triangles are similar by the Angle-Angle Similarity Postulate (AA ,).
If an angle of one triangle is congruent to an angle of a second triangle, and the
sides including the two angles are proportional, then the triangles are similar
by the Side-Angle-Side Similarity Theorem (SAS ,).
If the corresponding sides of two triangles are proportional, then the triangles are
similar by the Side-Side-Side Similarity Theorem (SSS ,).
Methods of indirect measurement use similar triangles and measurements to find
distances that are difficult to measure directly.
14. lM O lR, lN O lS,
MP
NP
lP O lT; MN
RS ≠ RT ≠ ST
14. If #MNP , #RST, which angles are congruent? Write an extended
proportion to indicate the proportional corresponding sides of
the triangles.
15. Art An artist is creating a stained glass window and wants it to be a
golden rectangle. To the nearest inch, what should be the length if the
width is 24 in.?
39 in. or 15 in.
The triangles are similar. Find the similarity ratio of the first to the second.
16. F 4 R D
46⬚
6
A
105⬚
8
17.
2:3
9
2.5 : 1 or 5 : 2
7.2
20
10
4
8
I
9
18
Y
x 2 Algebra The polygons are similar. Find the value of each variable.
x
18.
x ≠ 12; y ≠ 15
x
19.
2
9
6
20. nXYZ M nJKL;
SAS M Thm.
21. Not M ; Corr. sides
are not prop.
Z
408
6
3
8
Are the triangles similar? If so, write the similarity statement and name the
postulate or theorem you used. If not, explain.
20.
408
y
Chapter 7 Chapter Review
X
J
58º
58º
Y
L
21. Q
T
8
K
17
4 V
5
8.5
U
S
12
R
22. Two right triangles have an acute angle with the same measure. Name
the theorem or postulate that is the most direct way to prove the
triangles similar. AA M Post.
Alternative Assessment
Name
Class
L4
Date
Alternative Assessment
Form C
Chapter 8
TASK 1
23. Indirect Measurement A crate is 1.5 ft high and casts a 2-ft shadow.
At the same time, an apple tree casts an 18-ft shadow. How tall is
the tree? 13.5 ft
State and explain three ways to prove triangles similar. Include art in your
explanations.
TASK 2
C
A student claims that he has proven the following results for right
triangles. Evaluate each claim.
To find and use
relationships in similar
right triangles
When the altitude is drawn to the hypotenuse of a right triangle:
• the two triangles formed are similar to the original triangle and to each other;
• the length of the altitude is the geometric mean of the lengths of the segments
of the hypotenuse; and
• the length of each leg is the geometric mean of the length of the adjacent
hypotenuse segment and the length of the hypotenuse.
45ⴗ
G
F
A
D
E
B
2. The angle bisector of the right angle forms similar triangles
(䉭ACE ⬃ 䉭BCE).
© Pearson Education, Inc. All rights reserved.
7-4 Objective
The geometric mean of two positive numbers a and b is the positive number x such
that xa 5 xb.
1. The altitude to the hypotenuse forms similar triangles
(䉭ABC ⬃ 䉭ACD ⬃ 䉭CBD).
3. The midsegment connecting the legs forms a triangle similar to the
original (䉭CFG ⬃ 䉭CAB) with area one-half that of the original.
Geometry Chapter 8
Form C Test
29
x 2 Algebra Find the geometric mean of each pair of numbers.
24. 4 and 25 10
2!7
27. 19 and 28
3
25. 3 and 300 30
26. 5 and 12 2!15
28. 0.36 and 4 1.2
29. 2 !3 and !12
!12
x 2 Algebra Find the values of the variables. When an answer is not a whole number,
leave it in simplest radical form. x ≠ 2 "21; y ≠ 4 "3; z ≠ 4 "7
30.
31.
9
32.
x
x
y
16
y
To use the Side-Splitter
Theorem
To use the TriangleAngle-Bisector Theorem
y
4
x
8
14
z
x ≠ 15; y ≠ 12; z ≠ 20
7-5 Objectives
2
z
z
x ≠ 2 "3; y ≠ 6; z ≠ 4 "3
The Side-Splitter Theorem states that if a line is parallel to one side of a triangle
and intersects the other two sides, then it divides those sides proportionally. If
three parallel lines intersect two transversals, then the segments intercepted on the
transversals are proportional.
The Triangle-Angle-Bisector Theorem states that if a ray bisects an angle of a
triangle, then it divides the opposite side into two segments that are proportional
to the other two sides of the triangle.
x 2 Algebra Find the value of x.
33.
7.5
x
37.5
5.5
34.
35.
15
14
7
15
x
11
16
x
7
40
14
36.
12
15
37.
9
x
4
10
7
38.
12
16
x⫺3
x
x
11.25
17.5
12
Chapter 7 Chapter Review
409
409