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Name———————————————————————— Lesson
6.1
Date —————————————
Study Guide
For use with the lesson “Use Similar Polygons”
goal
Use proportions to identify similar polygons.
Vocabulary
Two polygons are similar polygons if corresponding angles are
congruent and corresponding side lengths are proportional.
If two polygons are similar, then the ratio of the lengths of two
corresponding sides is called the scale factor.
Theorem 1 Perimeters of Similar Polygons: If two polygons are
similar, then the ratio of their perimeters is equal to the ratios of their
corresponding side lengths.
example 1
Find the scale factor
Determine whether the polygons are
similar. If they are, write a similarity
statement and find the scale factor
of DEFG to JKLM.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Lesson 6.1
Corresponding Lengths in Similar Polygons: If two polygons are
similar, then the ratio of any two corresponding lengths in the polygons
is equal to the scale factor of the similar polygons.
J
D
15
G
12
K
12
E
6
F
9
20
M
16
8
L
Solution
From the diagram, you can see that ∠ D > ∠ J, ∠ E > ∠ K, ∠ F > ∠ L, and
∠ G > ∠ M. So, the corresponding angles are congruent.
DE
JK
15
20
3
4
EF
KL
6
8
3
4
3
4
12
16
FG
LM
GD
MJ
9
12
3
4
}
​   ​ 5 }
​   ​​ }  ​5 }
​   ​5 }
​   ​​  }  ​5 }
​   ​ 5 ​ } ​​ }  ​5 }
​    ​5 }
​   ​
​    ​5 }
The ratios are equal, so the corresponding side lengths are proportional.
3
So, DEFG , JKLM. The scale factor of DEFG to JKLM is }
​ 4 ​.
example 2
Use similar polygons
In the diagram, nABC , nPQR. Find the value of x.
Solution
BC
AB
}
​ PQ  ​ 5 }
​ QR  ​
14
6
P
A
Write proportion.
​ }
  ​5 }
​ 9 ​
x
Substitute.
6x 5 126
x 5 21
Cross Products Property
Solve for x.
14
15
x
B
6
22.5
C
9
R
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C6L01SG.indd 11
6-11
4/28/11 2:10:20 PM
Name———————————————————————— Lesson
6.1
Date —————————————
Study Guide continued
For use with the lesson “Use Similar Polygons”
Exercises for Examples 1 and 2
In the diagram, nRST , nXYZ.
45
1. Find the scale factor of nRST to nXYZ.
2. Find the value of a.
example 3
Y
S
R
50
27
T
a
X
Z
40
Find perimeters of similar figures
In the diagram, ABCD , FGHJ.
15
A
B
a. Find the scale factor of FGHJ to ABCD.
Lesson 6.1
30
8
12
b. Find the perimeter of FGHJ.
Solution
C
D
a. Because the figures are similar, the scale
J
H
10
factor is the ratio of corresponding sides.
8
G
F
8
2
FJ
​ }  ​ 5 }
​ 12  ​ 5 }
​ 3 ​
AD
example 4
Use a scale factor
X
In the diagram, nWXY , nFGH. }
Find the length of the altitude XZ​
​  .
G
Solution
25
W
First, find the scale factor of nWXY to nFGH.
Y
Z
18
WY
25
5
}
​ FH  ​5 }
​ 30 ​ 5 }
​ 6 ​
30
F
H
J
Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale
XZ
5
factor, you can write the proportion }
​ GJ  ​5 }
​ 6 ​. Then substitute 18 for GJ and solve for
}
XZ to find that the length of the altitude XZ​
​  is 15.
Exercises for Examples 3 and 4
In the diagram, LMNOP , RSTUV.
3.Find the scale factor of RSTUV to
LMNOP.
4. Find the perimeter of RSTUV.
}
5. Find the length of diagonal ​MO​ 
.
6-12
8
L
R
14
P
12
O
M
S
V
25
12
16
N
U
20
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
b. The perimeter of ABCD is 45. Let x be the perimeter of FGHJ. Using Theorem
2
x
6.1, you can write the proportion }
​ 45  ​5 }
​ 3 ​. So, the perimeter of FGHJ is x 5 30.
T
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C6L01SG.indd 12
4/28/11 2:10:20 PM
Answers for Chapter 6
Lesson 6.1 Use Similar
Polygons
Practice Level B
Teaching Guide
​ }
  ​ 5 }
​ FE  ​5 }
​ DE  ​  2. ∠ W > ∠ M, ∠ X > ∠ N,
DF
1. ∠ A > ∠ D, ∠ B > ∠ F, ∠ C > ∠ E,
AB
BC
AC
XY
WX
YZ
WZ
​ NO  ​ 5 }
​ OP  ​ 5 }
​ MP  ​
∠ Y > ∠ O, ∠ Z > ∠ P, }
​ MN  ​5 }
1
1
3. C 4. no 5. yes; BCDA , WXYZ; ​ } ​or
4
2
WXYZ , BCDA; 4
3
answers
1.
5
4
6. ​ } ​ 7. 15, 8, 135 8. 40 9. 50 10. }
​ 4 ​
5
4
11. m 5 11, n 5 4 12. m 5 8 13. 30 in., 21 in.
2
14. 9.75 in. 15. C 16. ​ } ​
5
17. XY 5 3.6, PN 5 15 18. 2.32
5
19. Area of n XYZ 5 6.96; Area of
nMNP 5 43.5; The ratio of the areas of
similar triangles equals the scale factor squared.
2. 5 pentagons, 20 obtuse isosceles triangles,
25 acute triangles and 1 kite
Practice Level A
1. ∠ A > ∠ D, ∠ B > ∠ E, ∠ C > ∠ F;
BC
AC
AB
​ }
  ​ 5 }
​ EF  ​5 }
​ DF  ​ 
DE
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Similarity
2. ∠ R > ∠ W, ∠ S > ∠ V, ∠ T > ∠ U;
RS
ST
RT
​ }
  ​ 5 }
​ VU  ​ 5 }
​ WU  ​ 
WV
3. ∠ C > ∠ M, ∠ D > ∠ J, ∠ E > ∠ K,
CD
CF
DE
EF
∠ F > ∠ L; }
​ MJ  ​5 ​ }
  ​5 }
​ KL  ​5 }
​ ML  ​ 
JK
4. ∠ P > ∠ Z, ∠ Q > ∠ W, ∠ R > ∠ X,
PQ
QR
RS
PS
∠ S > ∠ Y; ​ }
  ​5 }
​ 
  ​ 5 }
​ XY ​5 }
​ ZY  ​
ZW
WX
3
1
5. n ABC , n FDE; }
​ 2 ​ 6. GHIJ , KLMN; }
​ 2 ​
1
7. not similar 8. n ABC , n FDE; }
​ 1 ​ 9. 8
3
10. 18 11. ​ } ​; P(n LMN) 5 45, P(n PQR) 5 60
4
3
12. }
​ 5 ​; P(XYZW) 5 100, P(STUV) 5 60
3
20. }
​ 5 ​ 21. 60 ft 22. 3200 ft2
Practice Level C
1. ∠ S > ∠ C, ∠ T > ∠ D, ∠ U > ∠ E;
ST
TU
SU
​ }
  ​ 5 }
​ DE  ​5 }
​ CE  ​
CD
2. ∠ L > ∠ G, ∠ M > ∠ H, ∠ N > ∠ I;
MN
LN
LM
}
​ GH  ​5 }
​ HI  ​5 }
​ GI  ​
3. ∠ C > ∠ M, ∠ D > ∠ N, ∠ E > ∠ K,
CD
CF
DE
EF
∠ F > ∠ L; }
​ MN  ​ 5 }
​ NK  ​5 }
​ KL  ​5 }
​ ML  ​ 
4
4. n LNM , n TPO; }
​ 3 ​
5
5. quadrilateral ABCD , quadrilateral HEFG; }
​ 8 ​
3
2
6. ​ } ​ 7. ​ } ​ 8. 4.5 9. 1178 10. 24 11. 201.6 ft
3
2
rv
sv
tv
12. 77.4 in. 13. }
​ u  ​ 14. ​ }
  ​ 15. ​ }  ​ 16. 6
u
u
17. 25, 2.5 18. 8 19. 20.4, 15
20. 8 times greater
13. altitudes; x 5 24 14. medians; y 5 12
Study Guide
9
5
1. }
​ 10  ​  2. 36 3. }
​ 4 ​ 4. 77.5 5. 20
15. The shadow is not similar to the kite because
the corresponding side ratios are not all the same:
Problem Solving Workshop:
Mixed Problem Solving
95
1. a. 58,800 yen b. about 580.56 Canadian
2
dollars 2. a. ​ }7 ​, x 5 45.5, y 5 17.5 b. 50, 175
142
​ }
 ​ < 1.22 Þ 1.08 ø }
​ 132  ​
78
Geometry
Chapter Resource Book
A79