Feb 5, 2006 ... 10 ft. 8 ft. 6 ft. Reteach. Similar Figures and Proportions. 5-7. To determine if
AABC is similar to AXYZ, you can write a proportion for each pair of.
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Name
Date
Class
Reteach
LESSON
5-7 Similar Figures and Proportions Figures that have the same shape but not the same size are similar figures. In similar figures, the ratio of the lengths of the corresponding sides are proportional, and the corresponding angles have equal measures. To determine if 䉭ABC is similar to 䉭XYZ, you can write a proportion for each pair of corresponding sides.
A X 10 ft 6 ft Z 8 ft Y
15 ft
9 ft
longest sides
C 12 ft B
AB 15 3 XY 10 2
middle sides shortest sides BC 12 3 YZ 8 2
AC 9 3 XZ 6 2
The corresponding sides are always in 3
the ratio 2. So the triangles are similar. If a polygon has more than 3 sides, you must also show that the corresponding angles are equal. Identify the corresponding sides. Use ratios to determine whether the figures are similar. 1.
S 10 cm
T
16 cm
2.
D 4 cm 6 cm 12 cm E 8 cm F U
P 16 yd Q 24 yd
20 yd
F
12 yd
18 yd
R
16 2 SU TU ; EF 8 1 DF
12 2 ; 6 1
Are the triangles similar? 3.
J
54 m
K
no no
Are the triangles similar? 4.
L
yes
yes
13 in.
13 in. 13 in. 13 in. 13 in. 13 in. 13 in. 13 in.
not similar; anual
similar Copyright © by Holt, Rinehart and Winston. All rights reserved.
H
Are the ratios proportional?
X 24 m 18 m Z Y 72 m 12 m
36 m
15 yd
16 4 PR 24 4 PQ ; ; 12 3 FH 18 3 FG 20 4 QR 15 3 GH
10 5 ST 4 2 DE Are the ratios proportional?
G
measure 56
Holt Mathematics
MSM07G7_RESBK_Ch05_077-095.pe
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Reteach 5-7 Similar Figures and Proportions
Practice C 5-7 Similar Figures and Proportions
LESSON
LESSON
Figures that have the same shape but not the same size are similar figures. In similar figures, the ratio of the lengths of the corresponding sides are proportional, and the corresponding angles have equal measures.
Use the properties of similarity to determine whether the figures are similar. 1.
X Y
6.4 m
2.
R 1.6 m 2.4 m S 3.2 m T
4.8 m
3.2 m
Z
XY XZ YZ RS RT ST 3. 2 6.4 4.8 2 1.6 3.2 2.4 15 yd
W 12 yd X 10 yd Z
6 yd
M 8 yd
10 yd Y
P
DE EF 1.7 2.2 2.5 3.4 5.0 5.5 4.
N
14 cm
16 yd 15 yd O
H
ZY WZ WX not similar; MP MN NO XY 12 6 10 ; OP 16 8 15
E 14 cm F 104° 76° 14 cm 76° 104° G 14 cm
108°
A
108° L 21 cm
not similar; angles are not equal
15.6 in.
14.4 in.
B
?
6.
X 12 in. Y 5 in. 13 in. Z
15 m
D
10 m 48°
C
F
BC 6 in. 7.
10.5 cm 7.5 cm
7.5 cm
R 3.5 cm
S
19.5 m
F ? J
8.
G
10.5 cm H
FJ 4.9 cm
A 8.2 ft
Z
3.
54 m
K
F
12 yd
18 yd
H
Are the triangles similar? 4.
X 24 m 18 m Z Y 72 m 12 m
36 m
L
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Copyright © by Holt, Rinehart and Winston. All rights reserved.
G 15 yd
Are the ratios proportional?
no
yes
yes
13 in. 13 in. 13 in. 13 in. 13 in. 13 in. 13 in. 13 in.
not similar; angles not equal
similar
measure 56
Holt Mathematics
Problem Solving 5-7 Similar Figures and Proportions
Challenge 5-7 The Same, Only Bigger
LESSON
LESSON
Use the information in the table to solve problems 1–3.
You can sometimes create a similar figure by using copies of the original figure.
1. A small reproduction of one of the paintings in the list is similar in size. The reproduction measures 11 inches by 10 inches. Of which painting is this a reproduction?
Notice that the scale factor tells you how many times to repeat the original figure along each side or edge of the similar figure. Original Similar Figure Figure Scale factor = 2
J
4.6 ft
⬔C 90°, ⬔D 128° 55
20 yd
16 4 PR 24 4 PQ ; ; FG 12 3 FH 18 3 20 4 QR GH 15 3
no
Are the ratios proportional?
⬔A 68°, ⬔B 74°,
Copyright © by Holt, Rinehart and Winston. All rights reserved.
12 2 ; 6 1
10 5 DE 4 2
? D ?C 1.3 ft
74°
P 16 yd Q 24 yd
ST
4.1 ft B ? ?
4.3 ft
2.
U
Are the triangles similar?
68°
AC 9 3 XZ 6 2
D 4 cm 6 cm E 8 cm F
16 2 SU TU ; EF 8 1 DF
N
Y 8.6 ft X 128° 2.6 ft 90° W 9.2 ft
9.1 cm
shortest sides
R
13 m ?
?
12 cm 16 cm
L ?
⬔M 29°, ⬔N 103°, ⬔L 48°
Q 6.5 cm T
T
26 m
20 m
M
S 10 cm
29° 103°
middle sides BC 12 3 YZ 8 2
Identify the corresponding sides. Use ratios to determine whether the figures are similar. 1.
E
longest sides AB 15 3 XY 10 2
If a polygon has more than 3 sides, you must also show that the corresponding angles are equal.
The figures in each pair are similar. Find the missing lengths or angle measures. 5.
C 12 ft B
The corresponding sides are always in 3 the ratio 2. So the triangles are similar.
21 cm 72°
15 ft
9 ft
21 cm K 72°
21 cm
M
X 10 ft 6 ft Z 8 ft Y
DF
J
To determine if 䉭ABC is similar to 䉭XYZ, you can write a proportion for each pair of corresponding sides.
A
BC AB AC not similar; ;
similar; ;
3.
D 3.4 ft E A 2.2 ft 2.5 ft 5.5 ft 5.0 ft B C 1.7 ft F
The Dance Class
Original Similar Figure Figure Scale factor = 3
Use the given scale factor and copies of the original figure to draw a figure similar to the original figure. 1.
2. A local artist painted a reproduction of Cézanne’s painting. It measures 88 inches by 72 inches. Is the reproduction similar to the original? What is the ratio of corresponding sides?
2.
Artist
Original Size (in.)
Mona Lisa
Leonardo da Vinci
30 by 21
The Dance Class
Edgar Degas
33 by 30
The Blue Vase
Paul Cézanne
22 by 18
3. A poster company made a poster of da Vinci’s painting. The poster is 5 feet long and 3.5 feet wide. Is the poster similar to the original Mona Lisa? What is the ratio of corresponding sides?
yes; 1:4
scale factor 2
scale factor 4
Painting
yes; 1:2
Choose the letter for the best answer.
3.
4.
scale factor 2
scale factor 3
5. Draw a figure in the space below. Use a scale factor of 2 to create a similar figure. Drawings will vary. Possible drawing
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57
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given.
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4. Triangle ABC has sides of 15 cm, 20 cm, and 25 cm. Which triangle could be similar to triangle ABC? A A triangle with sides of 3 cm, 嘷 4 cm, and 5 cm B A triangle with sides of 5 cm, 6 cm, and 8 cm C A triangle with sides of 30 cm, 40 cm, and 55 cm D A triangle with sides of 5 cm, 10 cm, and 30 cm
5. A rectangular picture frame is 14 inches long and 4 inches wide. Which dimensions could a similar picture frame have? F Length = 21 in.; width = 8 in. G Length = 35 in.; width = 15 in. H Length = 49 in.; width = 14 in. 嘷 J Length = 7 in.; width = 3 in.
6. A rectangle is 12 meters long and 21 meters wide. Which dimensions correspond to a nonsimilar rectangle? A 4 m; 7 m C 20 m; 35 m D 24 m; 35 m 嘷 B 8 m; 14 m
7. A rectangle is 6 feet long and 15 feet wide. Which dimensions correspond to a similar rectiangle? F 8 ft; 24 ft H 15 ft; 35 ft J 18 ft; 40 ft G 10 ft; 25 ft 嘷
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90
58
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Holt Mathematics