ratios and similar polygons - learning resource center
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Ratios and Similar Polygons
A useful way to compare two quantities is write a ratio. If your family has 2 dogs and 3 cats, the ratio of dogs to cats is 2/3 or 2:3.
Must be
the same
units!!
An equation that equates two
ratios is a proportion.
Properties of Proportion
Properties of Proportion
Solving Proportions
Solving Proportions
Solution: 4x = 5*8x = 10
Solve the Proportion
Solve the Proportion
Solution: 4(x+5) = 16(x+2)4x+20 = 16x + 32
-12 = 12 xx = -1
AB
CD
CF
AB
AB
CD
CF
AB
Solution:
2 or 14 2
92
Solve for x
Solve for x
Solution: 2 = x5 20
2*20 = 5x
8 = x
The Area of a rectangle is 108. The
ratio of length to width is 3:4.
What is the length and width of the
rectangle?
l
w
The Area of a rectangle is 108. The
ratio of length to width is 3:4.
What is the length and width of the
rectangle?
l
wSolution:
3x*4x = 10812x^2 = 108
x^2 = 9 l = 3x = 3*3 = 9x = 3 w = 4x = 4*3 = 12
The measures of the angles in a
triangle are in the extended ratio
1:4:7. Find the measure of each
angle.
The measures of the angles in a triangle are in
the extended ratio 1:4:7. Find the measure of
each angle.
Solution: 1x+4x+7x = 180
12x = 108x = 9
x = 9o 4x = 36o 7x = 63o
Similar Polygons
• When there is a correspondence between two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional….
• The two polygons are called Similar Polygons
Quad ABCD ~Quad EFGH
• ~ means Similar
• List all the the pairs of congruent angles.
• Write the ratios of the corresponding sides in a statement of proportionality
Quad ABCD ~Quad EFGH• ~ means Similar
• List all the the pairs of congruent angles.
<A~ <E <B~ <F <D~ <H <C~ <G
• Write the ratios of the corresponding sides in a statement of proportionality
AB = BC = CD = AD
EF FG GH EH
ABC DEF • List all congruent
angles.
• Write the ratios of corresponding sides.
ABC DEF • List all congruent angles.
<A~ <D <B~ <E <C~ <F
• Write the ratios of corresponding sides.
AB = BC = AC
DE EF DF
Theorem
If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding lengths.
Definition
If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the Scale Factor
Pentagon ABCDE ~ JKLMN
• What is the scale factor?
• What is the value of x?
Pentagon ABCDE ~ JKLMN
• What is the scale factor?
5/6
• What is the value of x?
5 = 4
6 x
5x = 4*6
x = 24/5 = 4.8
Are these Triangles similar?
• If so, write a similarity statement.
Are these Triangles similar?
• If so, write a similarity statement.
Solution:
2 = 3 = 4
4 6 8
Yes
ABC ~ XYZ
Decide whether the figures are similar.
If they are, write the similarity statement.
Decide whether the figures are similar.
If they are, write the similarity statement.
Solution:
4 = 9 = 12
10.5 12 18
NO
You have a photo 4 in. wide by 6 in. long
that you want to reduce to fit in a frame
that is 1.5 in. wide. How long will the
reduced photo be?
4 in
1.5 in
6 in
x
You have a photo 4 in. wide by 6 in. long that
you want to reduce to fit in a frame that is 1.5
in. wide. How long will the reduced photo be?
Solution: 4 = 1.56 l 4l = 9 l = 2.25 in.
4 in
1.5 in
6 in
x
A painting is similar to the wall on which
it is hanging. Calculate the scale factor of
the wall to the painting and find the ratio
of their perimeters.
A painting is similar to the wall on which
it is hanging. Calculate the scale factor of
the wall to the painting and find the ratio
of their perimeters.
Solution:
3.2 = 6 = 2 Ratio of perimeters is the same.8 15 5
Parallelogram ABCD is similar to
parallelogram GBEF.
Find the value of y.
B
E
C
F
D
G A
15
24
12
y
Parallelogram ABCD is similar to
parallelogram GBEF. Find the value of y.
B
E
C
F
D
G A
15
24
12
y
Solution:
12 = 15y 24 12 * 24 = 15y y = 19.2
ΔUVW is similar to
ΔYXW.
Find the value of a.
ΔUVW is similar to
ΔYXW.
Find the value of a.
Solution:
12 = 6.4a 5 12 * 5 = 6.4a a = 9.4
Solution:
5 = 12 5(x – 1) = 10 *1210 x - 1 5x – 5 = 120 x = 25
5 = 13 5(y + 20) = 10 *1310 y + 20 5y + 100 = 130 y = 6
Sometimes, Always or Never?
• Two isosceles triangles are similar
• Isosceles trapezoids are similar
• Squares are similar
• Equilateral triangles are similar
• An isosceles triangle and a scalene triangle are similar.
Sometimes, Always or Never?
• Two isosceles triangles are similar
Sometimes
• Isosceles trapezoids are similar
Sometimes
• Squares are similar
Always
• Equilateral triangles are similar
Always
• An isosceles triangle and a scalene triangle are similar.
Never