Ratios and Similar Polygons - Learning Resource Center
Transcript of Ratios and Similar Polygons - Learning Resource Center
![Page 1: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/1.jpg)
Ratios and Similar Polygons
![Page 2: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/2.jpg)
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.
![Page 3: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/3.jpg)
Must be
the same
units!!
![Page 4: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/4.jpg)
An equation that equates two
ratios is a proportion.
![Page 5: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/5.jpg)
Properties of Proportion
![Page 6: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/6.jpg)
Properties of Proportion
![Page 7: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/7.jpg)
Solving Proportions
![Page 8: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/8.jpg)
Solving Proportions
Solution: 4x = 5*8x = 10
![Page 9: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/9.jpg)
Solve the Proportion
![Page 10: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/10.jpg)
Solve the Proportion
Solution: 4(x+5) = 16(x+2)4x+20 = 16x + 32
-12 = 12 xx = -1
![Page 11: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/11.jpg)
AB
CD
CF
AB
![Page 12: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/12.jpg)
AB
CD
CF
AB
Solution:
2 or 14 2
92
![Page 13: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/13.jpg)
Solve for x
![Page 14: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/14.jpg)
Solve for x
Solution: 2 = x5 20
2*20 = 5x
8 = x
![Page 15: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/15.jpg)
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
![Page 16: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/16.jpg)
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
![Page 17: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/17.jpg)
The measures of the angles in a
triangle are in the extended ratio
1:4:7. Find the measure of each
angle.
![Page 18: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/18.jpg)
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
![Page 19: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/19.jpg)
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
![Page 20: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/20.jpg)
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
![Page 21: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/21.jpg)
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
![Page 22: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/22.jpg)
ABC DEF • List all congruent
angles.
• Write the ratios of corresponding sides.
![Page 23: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/23.jpg)
ABC DEF • List all congruent angles.
<A~ <D <B~ <E <C~ <F
• Write the ratios of corresponding sides.
AB = BC = AC
DE EF DF
![Page 24: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/24.jpg)
Theorem
If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding lengths.
![Page 25: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/25.jpg)
Definition
If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the Scale Factor
![Page 26: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/26.jpg)
Pentagon ABCDE ~ JKLMN
• What is the scale factor?
• What is the value of x?
![Page 27: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/27.jpg)
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
![Page 28: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/28.jpg)
Are these Triangles similar?
• If so, write a similarity statement.
![Page 29: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/29.jpg)
Are these Triangles similar?
• If so, write a similarity statement.
Solution:
2 = 3 = 4
4 6 8
Yes
ABC ~ XYZ
![Page 30: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/30.jpg)
Decide whether the figures are similar.
If they are, write the similarity statement.
![Page 31: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/31.jpg)
Decide whether the figures are similar.
If they are, write the similarity statement.
Solution:
4 = 9 = 12
10.5 12 18
NO
![Page 32: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/32.jpg)
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
![Page 33: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/33.jpg)
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
![Page 34: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/34.jpg)
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.
![Page 35: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/35.jpg)
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
![Page 36: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/36.jpg)
Parallelogram ABCD is similar to
parallelogram GBEF.
Find the value of y.
B
E
C
F
D
G A
15
24
12
y
![Page 37: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/37.jpg)
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
![Page 38: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/38.jpg)
ΔUVW is similar to
ΔYXW.
Find the value of a.
![Page 39: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/39.jpg)
ΔUVW is similar to
ΔYXW.
Find the value of a.
Solution:
12 = 6.4a 5 12 * 5 = 6.4a a = 9.4
![Page 40: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/40.jpg)
![Page 41: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/41.jpg)
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
![Page 42: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/42.jpg)
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.
![Page 43: Ratios and Similar Polygons - Learning Resource Center](https://reader031.fdocuments.us/reader031/viewer/2022012502/617ba7b4137fcf06de333939/html5/thumbnails/43.jpg)
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