Embed Size (px)
Citation preview
Geometry
8.3 Similar Polygons
April 21, 2023 Geometry 8.3 Similar Polygons 2
Goals
Identify similar polygons Find the ratio of similarity between similar
figures. Solve problems involving similar figures.
April 21, 2023 Geometry 8.3 Similar Polygons 3
Similar Polygons
ABCD and RSTV are similar polygons: Corresponding angles are congruent. Corresponding sides are proportional.
RS
T
V
108
6A
B
C
D
15
12
9
96
April 21, 2023 Geometry 8.3 Similar Polygons 4
Similar Polygons
Corresponding angles are congruent: A R, B S, C T, D V
RS
T
V
108
6A
B
C
D
15
12
9
96
April 21, 2023 Geometry 8.3 Similar Polygons 5
Similar Polygons
Corresponding sides are proportional:
RS
T
V
108
6A
B
C
D
15
12
9
9
15 12 9 9
10 8 6 63 3 3 3
2 2 2 2
6
April 21, 2023 Geometry 8.3 Similar Polygons 6
Similar Polygons Corr. s Sides prop. ABCD ~ RSTV
RS
T
V
108
6A
B
C
D
15
12
9
96
April 21, 2023 Geometry 8.3 Similar Polygons 7
Similar Polygons
Corresponding Angles are congruent. Corresponding Sides are proportional. Use the symbol “~” for similar.
To show that two polygons are similar, you must prove both things: angles congruent, sides proportional.
April 21, 2023 Geometry 8.3 Similar Polygons 8
Similarity Statements
List the congruent angles.
Write the ratios of the corresponding sides.
7070
J
K
L Q R
S
JK JL KL
QS QR SR
J Q, K S, L R
April 21, 2023 Geometry 8.3 Similar Polygons 9
Example
Are these figures similar? Yes Why? Corr. angles congruent Corr. sides proportional.
E
N
F
GH
M
O P
3
2
4
1.5
6
8
34
April 21, 2023 Geometry 8.3 Similar Polygons 10
Write the similarity statements.
E
N
F
GH
M
O P
E N, F M
G P, H O
EF FG GH HE =
NM MP PO ON
EFGH ~ NMPO
Or: HEFG ~ ONMP, GFEH ~ PMNO, EHGF ~ NOPM, etc.
3
2
4
1.5
6
8
34
April 21, 2023 Geometry 8.3 Similar Polygons 11
Scale Factor
If two polygons are similar, the ratio of two corresponding sides is called the scale factor.
April 21, 2023 Geometry 8.3 Similar Polygons 12
Scale Factor
The scale factor of JKL to QSR is 10/5 or 2/1.
The scale factor of QSR to JKL is 5/10 or 1/2.
7070
J
K
L Q R
S10 5
April 21, 2023 Geometry 8.3 Similar Polygons 13
Perimeter and Similar Figures
A B
CD
F G
HI
10
4
1414
5
z
yx
1. ABCD ~ FGHI
2. Find the scale factor of ABCD to FGHI.
3. Find the values of x, y, and z.
4. Find the perimeter of ABCD and FGHI.
5. Find the ratio of the perimeters.
April 21, 2023 Geometry 8.3 Similar Polygons 14
Perimeter and Similar Figures
A B
CD
F G
HI
10
4
1414
5
z
yx
2. Find the scale factor from ABCD to FGHI.
The only known corresponding sides are AB and FG. 10 2
5 1
April 21, 2023 Geometry 8.3 Similar Polygons 15
Perimeter and Similar Figures
10 14
510 70
7
xx
x
2 14
1
2 14
7
y
y
y
A B
CD
F G
HI
10
4
1414
5
z
yx
3. Find the values of x, y, and z.
2 4
12 4
2
zz
z
April 21, 2023 Geometry 8.3 Similar Polygons 16
Perimeter and Similar Figures
A B
CD
F G
HI
10
4
1414
5
2
77
4. Find the perimeter of ABCD and FGHI.
P = 42P = 21
April 21, 2023 Geometry 8.3 Similar Polygons 17
Perimeter and Similar Figures
A B
CD
F G
HI
10
4
1414
5
2
77
5. Find the ratio of the perimeters.
P = 42P = 21
Ratio of perimeters 2:1
Ratio of Similarity 2:1
April 21, 2023 Geometry 8.3 Similar Polygons 18
Theorem 8.1
If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths and is equal to the similarity ratio.
April 21, 2023 Geometry 8.3 Similar Polygons 19
Theorem 8.1 Example
20
8
P = 100 P = ?
These figures are similar. Find the perimeter of the smaller one.
20 100
820 800
40
PP
P
April 21, 2023 Geometry 8.3 Similar Polygons 20
Problems to Solve
April 21, 2023 Geometry 8.3 Similar Polygons 21
Problem 1
20
48
6y – 2
x + 6
Solve for x and y if the triangles are similar.
April 21, 2023 Geometry 8.3 Similar Polygons 22
Problem 1 Solution
20 6
8 48( 6) 80
6 10
4
x
x
x
x
20 2
8 68( 2) 120
2 15
17
y
y
y
y
20
48
6y – 2
x + 6
Scale Factor is 20/8
Solve
for x
Solve
for y
April 21, 2023 Geometry 8.3 Similar Polygons 23
Problem 2
60
x + 10
95
85100
y24
32
Find x and y if the figures are similar.
April 21, 2023 Geometry 8.3 Similar Polygons 24
Problem 2 Solution
60 10
24 321920 24 240
1680 24
70
x
x
x
x
60
x + 10
95
85100
y24
32
Similarity Ratio
y = 360 - 100 - 85 - 95
y = 80
April 21, 2023 Geometry 8.3 Similar Polygons 25
Problem 3
ABC ~ RST AB = 20 ST = 4 BC = RS Find BC and RS.
April 21, 2023 Geometry 8.3 Similar Polygons 26
Problem 3 Solution
ABC ~ RST AB = 20 ST = 4 BC = RS Find BC and RS.
A
B C
R
S T
20
4x
x
April 21, 2023 Geometry 8.3 Similar Polygons 27
Problem 3 SolutionA
B C
R
S T
20
4x
x2
20
4
80
80 16 5
4 5
8.94
AB BC
RS STx
x
x
x
x
x
ABC ~ RST
April 21, 2023 Geometry 8.3 Similar Polygons 28
Problem 4
You want to print a picture in your camera. You have two sizes of paper for your printer: 4 × 6 and 5 × 7. Does it matter? Will the pictures printed from each size of paper be similar?
4 × 6
5 × 7
4 6
5 728 30
Sides not proportional, figures not similar.
April 21, 2023 Geometry 8.3 Similar Polygons 29
Problem 5
MNOP has a perimeter of 24. Find the perimeter of QRST if MN = 8 and QR = 12.
8
12
8 24
128 288
36
MN
QR
PP
P
April 21, 2023 Geometry 8.3 Similar Polygons 30
Summary
Two polygons are similar if they have the same shape, but a different size.
If polygons are similar corresponding angles are congruent, and corresponding sides are proportional.
The ratio of any two corresponding sides is the scale factor.
The ratio of the perimeters is equal to the ratio of two corresponding sides.
