1. Ch 7 Goals and common core standards Ms. Helgeson
Find and simplify the ratio of two numbers.
Use properties of proportions.
Identify similar polygons.
Identify similar triangles.
Use similarity theorems to prove that two triangles are similar.
Use proportionality thms to calculate segment length
Identify dilations.

2. CC.9-12.G.SRT.2
CC.9-12.G.SRT.3
CC.9-12.G.SRT.4
CC.9-12.G.SRT.5
CC.9-12.G.CO.2

3. OUTCOMES IN Topic 7 THE STUDENT SHOULD BE ABLE TO:
RATIO AND PROPORTION
Goal 1: Find and simplify the ratio of two numbers.
Goal 2: Use proportions to solve real-life problems.
PROBLEM SOLVING IN GEOMETRY WITH PROPORTIONS

4. Goal 1: use properties of proportions.
Goal 2: use proportions to solve real-life problems.
SIMILAR POLYGONS
Goal 1: identify similar polygons.
Common Core Standard:
CC.9-12.G.SRT.2 &
CC.9-12.G.CO.2
SIMILAR TRIANGLES
Goal: identify similar triangles.

5. CC.9-12.G.SRT.3
CC.9-12.G.SRT.5
PROVING TRIANGLES ARE SIMILAR
Goal: use similarity theorems to prove that two triangles are similar.
Common Core Standard:

6. Common Core Standard:
CC.9-12.G.SRT.2
CC.9-12.G.CO.2

7. PROPORTIONS AND SIMILAR TRIANGLES
Goal: use proportionality theorems to calculate segment lengths.
Common Core Standard:
CC.9-12.G.SRT.4
DILATIONS
Goal: identify dilations.

8. Topic 7
Similarity

9. Real World Uses of Congruence and Similarity

10. Similarity PANTOGRAPH RECIPES ARCHITECTURAL DRAWINGS MAPS GOLDEN RATIO
MAPS
Similarity
RECIPES
PANTOGRAPH
ARCHITECTURAL DRAWINGS
HEIGHTS OF TREES, BUILDINGS, ETC
SHADDOW PUPPETS
MODELS FOR HOUSES, BOATS, CARS, ETC
PLAYING TENNIS, PING PONG, POOL

11. Ratio and Proportion Ratio—a comparison of two quantities.
The ratio of “a” to “b” can be expressed as
a/b, where b ≠ 0.
The ratio can also be written as a:b

12. Write the indicated ratio for the members in your class today
Write the indicated ratio for the members in your class today. Then create some ratios of your own about your class.
Boys : girls
girls : boys
left-handed : right-handed
Have no pet at home : have at least 1pet

13. Express the ratio as a fraction in simplest form.
2 inches on a map represents 150 miles
inch
miles =

14. Converting Units
12 inches = 1 foot
3 feet = 1 yard
5280 feet = 1 mile = 1760 yards
16 ounces = 1 pound

15. Metric System

16. Simplifying ratios with unlike units:
convert to like units so that the units divide out. Then simplify the fraction, if possible.
1) 12 cm 2) 6 ft 3) 1 m 4) 3 yd
4 m in km ft
1m = 100cm
1ft = 12 in
1 km = 1000m
1 yd = 3 ft

17. A rectangular field has a length of one kilometer and a width of 300 meters. Find the ratio of the length to the width. Remember: 1 km = 1000 m 10:3 A telephone pole 7 meters tall snaps into two parts during a wind storm. The ratio of the two parts is 3 to 2. Find the length of each part. 4.2 m, 2.8 m

18. Three numbers aren’t known, but the ratio of the numbers is 1:2:5
Three numbers aren’t known, but the ratio of the numbers is 1:2:5. Is it possible that the numbers are 1, 2, and 5? 10, 20, and 50? 3, 6, and 20? X, 2x, and 5x?

19. Using Ratios
The perimeter of rectangle ABCD is 60 cm. The ratio of AB:BC is 3:2. Find the length and width of the rectangle. C D A B

20. 1. The perimeter of the isosceles triangle shown is 56 in
1. The perimeter of the isosceles triangle shown is 56 in. The ratio of LM:MN is 5:4. Find the lengths of the sides and the base of the triangle. L N M

21. 1. The measure of the angles in ∆JKL are the extended ratio of 1:2:3
1.The measure of the angles in ∆JKL are the extended ratio of 1:2:3. Find the measures of the angles.
2. The ratios of the side lengths of ∆DEF to the corresponding side lengths of ∆ABC are 2:1. Find the unknown lengths. C F
3 in B A D E
8 in

22. Proportion An equation that equates two ratios is a proportion. a c
b d =

23. Properties of Proportions
Cross Product Property: The product of the extremes equals the product of the means.
If a c , then ad = bc
b d
Ex: 5/7 = 25/35
5 x 35 = 7 x 25 =

24. Reciprocal Property: If two ratios are equal, then their reciprocals are also equal.
If a c , then b d
b d a c = =

25. Examples: Solve the prop.
3/(y + 2) = 2/y
(3x – 1)/4 = 7/8
(x + 3)/8 = 4/3
If one U.S. dollar is worth 0.61 British Pound Sterling, and a shirt cost $19.95 in U.S. currency, how much would it cost in Great Britain? £12.23

26. The ratio of the measures of two consecutive angles between the parallel sides of a trapezoid is 5:7. The ratio of the other two consecutive angle measures between the parallel sides is 1:3. Find the measures of the angles in the trapezoid.

27. 8.2 Problem Solving in Geometry with Proportions

28. Additional Properties of Proportions.
If a/b = c/d, then a/c = b/d.
If a/b = c/d, then (a + b)/b = (c + d)/d

29. True or False

30. 5. If p/6 = r/10, then p/r = 3/5
6. If a/3 = c/4, then (a+3)/3 = (c+3)/4

31. In the diagram MQ/MN = LQ/LP. Find the length of LQ. 6 N 15 13 L 5 P Q

32. The geometric mean of two positive numbers a and b is the positive number x such that a/x = x/b.
Ex: What is the geometric mean of 8 and
18?

33. Use a calculator to complete the table for the proportion a/x = x/b.5 20 ? 8 4 2

34. Using Proportions in Real Life
A scale model of the Titanic is inches long and inches wide. The Titanic itself was feet long. How wide was it in feet?
92.38 ft
width
length

36. 7.1 Dilations
Pupils dilate either larger or smaller depending on the amount of light that enters the eye.  This real world example helps us to understand the use of dilation in geometry as well – dilation is a transformation that produces an image that is the same shape as the pre-image but is a different size, either larger or smaller.

37. Definition
A dilation with center O and scale factor k is a transformation that maps every point A in the plane to a point A’ so that the following properties are true:
Next page

38. If A is the center point O, then A = A’.
1. If A is not the center point O, then the image point A’ lies on OA. The scale factor k is a positive number such that
k = OA’ / OA and k ≠ 1.
If A is the center point O, then A = A’. A A´ A´ O A
Reduction O
Enlargement
O = A = A´
Distance is not preserved!

39. enlargement reduction
The dilation is a reduction if 0 < k < 1 (proper fraction) and it is an enlargement if k > 1.
k = OA´/OA
enlargement reduction
K = 3/6 = 1/2
K = 5/2 6 5 3 2 O

40. Identify the dilation and find its scale factor. a) (b)3 7 A Q S T 1 B A’ R’ D Q’ S’ T’ B’ 3 D’ C
Reduction or Enlargement?
Scale factor?
Reduction or Enlargement?
Scale factor?

41. Answer to a: Enlargement, k = 7/3
Answer to b: Reduction, k = 3/4

42. Dilation is NOT an isometric transformation so its properties differ from the ones we saw with reflection, rotation and translation. The following properties are preserved between the pre-image and its image when dilating:
Angle measure  (angles stay the same)
Parallelism (things that were parallel are still parallel)
Collinearity (points on a line, remain on the line)

48. D (BA)
1/2

49. Dilation in a Coordinate Plane
Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4). Use the origin as the center and use a scale factor of ½. How does the perimeter of the preimage compare to the perimeter of the image?
Same as: D (ABCD)
(1/2, O)

50. Solution:
Because the center of the dilation is the origin, you can find the image of each vertex by multiplying its coordinates by the scale factor.
A(2, 2) →A’(1, 1)
B(6, 2) →B’(3, 1)
C(6, 4) →C’(3, 2)
D(2, 4) →D’(1, 2)
From the graph, you can see that the preimage has a perimeter of 12 and the image has a perimeter of 6. A preimage and its image after a dilation are similar figures. Therefore, the ratio of the perimeters of a preimage and its image is equal to the scale factor of the dilation.

51. A dilation of 1/2 with the center of dilation at the origin. D

52. D (TH)
(2, T)

53. D (GT)
(1/4, T)

54. Ex: Dilation of 2 with the center of dilation at (6, 4).

55. Ex: A dilation of ½ with the center of dilation at (-6, 2).

56. Page 507

57. Shadow puppets
Pilobolus.wmv

59. 7.2 Similarity Transformations
Graph a Composition of a Rigid Motion and a Dilation.
The vertices of ∆XYZ are X(3, 5), Y(-1, 4), and Z(1, 7).
What is the graph of the image
(D ◦ T )(∆XYZ)? (Center at origin when not
specified.)
(D ◦ r )(∆XYZ)? 2
<1, -2> 3
(90˚, O)

60. Is there a composition of transformations that maps ∆XYZ to ∆JKL
Is there a composition of transformations that maps ∆XYZ to ∆JKL? Explain.
Rotation 180˚ about the origin;
Dilation with scale factor of 2.
D ◦ r maps
∆XYZ to ∆JKL. Y 2
(180˚, O) X Z J L K

61. Similarity Transformation
A similarity transformation is a composition of one or more rigid motions and a dilation. A similarity transformation results in an image that is similar to the preimage. A composition of a reflection, a translation, and a dilation is a similarity transformation. A similarity transformation is a composition of one or more rigid motions and a dilation. All circles are similar to each other.

63. 7.3 Similar Polygons and Proving Triangles Similar
1.When figures have the same shape but different sizes, they are called similar figures.
2.Def. of Similar Polygons—Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.

64. D 10 H 5 A E 8 4 5
2.5 F G C 3 B 6
Quad ABCD Quad EFGH

65. The ratio of the lengths of two corresponding sides of two similar polygons is called the scale factor. #2 #1 5 10
Scale Factor #1 to #2 is 2 to 1

66. Congruent Figures
In order to be congruent, two figures must be the same size and same shape.

67. Similar Figures
Similar figures must be the same shape, but their sizes may be different.

68. Similar Figures
This is the symbol that means “similar.”
These figures are the same shape but different sizes.

69. SIZES
Although the size of the two shapes can be different, the sizes of the two shapes must differ by a factor. 4 2 6 6 3 3 1 2

70. When you have a photograph enlarged, you make a similar photograph.
Enlargements
When you have a photograph enlarged, you make a similar photograph.
X 3

71. Reductions
A photograph can also be shrunk to produce a slide. 4

72. Determine the length of the unknown side.15 12 ? 4 3 9

73. When changing the size of a figure, will the angles of the figure also change?40 70 ? ? 70

74. Nope! Remember, the sum of all 3 angles in a triangle MUST add to 180 degrees. If the size of the angles were increased, the sum would exceed 180 degrees. 40 40 70 70 70 70

75. We can verify this fact by placing the smaller triangle inside the larger triangle.40 40 70 70 70 70

76. The 40 degree angles are congruent.70 70 70 70

77. The 70 degree angles are congruent.40 40 70 70 70 70 70

78. The other 70 degree angles are congruent.4 40 70 70 70 70 70

79. Find the length of the missing side.50 ? 30 6 40 8

80. This looks messy. Let’s translate the two triangles.50 ? 30 6 40 8

81. Now “things” are easier to see.50 30 ? 6 40 8

82. Similarity is used to answer real life questions.
Suppose that you wanted to find the height of this tree.

83. You can measure the length of the tree’s shadow.
10 feet

84. Then, measure the length of your shadow.
10 feet
2 feet

85. If you know how tall you are, then you can determine how tall the tree is.
6 ft
10 feet
2 feet

86. The tree must be 30 ft tall. Boy, that’s a tall tree!
10 feet
2 feet

87. Similar Figures
So, similar figures are two figures that are the same shape and whose sides are proportional.

88. 1) Determine the missing side of the triangle.? 9 5 3 4 12

89. 4) Determine the height of the lighthouse.? 8
2.5 10

90. Write three equal ratios to show corresponding sides are proportional.
∆ABC ∆DEF
Write three equal ratios to show corresponding sides are proportional.
Find the scale factor of ∆ABC to ∆DEF.
Find the value of x and y.
Find the ratio m<A / m<D.
Find the perimeter of ∆DEF.
What is ratio of perm. Of ∆ABC & ∆DEF? E B 9 x 8 y F D 16 12 A C

91. Write four equal ratios to show corresponding sides are proportional.
Quad ABCD Quad EFGH
Write four equal ratios to show corresponding sides are proportional.
Find the scale factor.
Find AB, HG, and FG.
Find the perimeter of both Quad.
What is the ratio of Quad ABCD and Quad EFGH. 4 C B G F 2 5 E H 6 A D 8

92. Similar Triangles In the diagram, ∆BTW ∆ETC.
Write the statement of proportionality.
Find m<TEC.
Find ET and BE. T
34° E C 3 20 79 W B 12

93. AA Similarity Postulate
If two-angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

94. Ex: Given: AB parallel to CD, AB = 4,
AE = 3x + 4, CD = 8, ED = x + 12.
Find AE and DE. C A E B D

95. Given: AB parallel to DE, DA = 2, CA = 8, CE = 3 Find: CB.

96. Using a Pantograph to resize the drawing.

97. P
Suction cup
brads R
Tracing pin
As you move the tracing pin of a pantograph along a figure, the pencil T Q T
Attached to the far end draws an enlargement. As the pantograph expands and contracts, the three brads and the tracing pin always form the vertices of a parallelogram. The ratio of PR to PT is always equal to the ratio of PQ to PS. Also, the suction cup, the tracing pin, and the pencil remain collinear. S
4.8 in

98. SSS Similarity Theorem: If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar. 4 5 8 10 3 6

99. How high is the cliff? D B
5 ft
6.5 ft
200 ft A C E

100. SAS Similarity Thm
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

101. To estimate the height of a tree, a scout sights the top of the tree in a mirror that is 34.5 m from the tree. The mirror is on the ground and faces upward. The scout is 0.75 m from the mirror, and the distance from his eyes to the ground is about 1.75m. How tall is the tree? x
1.75 m
34.5 m
0.75 m

103. 7.4 Similar Right Triangles
Thm 9.1: If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. ΔCBD ~ ΔABC, ΔACD ~ ΔABC, and ΔCBD ~ ΔACD C A B D

104. The measures of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse. AD CD CD BD
The geometric mean is the number x such that
a/x = x/b,
Where a, b, and x are positive numbers. = C A B D

105. If the altitude is drawn to the hypotenuse of a right triangle, then the measure of a leg of the triangle is the geometric mean between the measures of the hypotenuse and the segment of the hypotenuse adjacent to that leg. AD AC BD BC AC AB BC AB = = C B A D

106. Instead of memorizing Theorem 9
Instead of memorizing Theorem 9.2, you could use the principle of similarity. To find h, set up a proportion between the 2 smaller triangles since they are similar. The ratio of the short leg to the long leg is a constant. short leg 3 h long leg h 7
1st ∆
2nd ∆ = = a b h
1st
2nd 7 3

107. You could also set up a proportion between one of the two smaller triangles and the larger triangle. 1st ∆ largest triangle short leg 3 a hypotenuse a 10
Ex: 1 = = b a h
1st 3 7 10

108. Ex: 2 long leg 7 b hyp b 10
2nd ∆
Largest ∆ = = b a h
2nd ∆ 7 3 10

109. Two smaller ∆’s are similar
Two smaller ∆’s are similar. short leg 3 h long leg h 7 Small ∆ & largest ∆: similar. short leg 3 a hyp a 10 Larger ∆ & largest ∆: similar long leg 7 b hyp b 10 a b h 3 7 = = =

110. Examples AD = 4, BD = 9, CD = ___ CD = 8, BD = 16, AD = ____
AD = 7, AB = 11, CD = ___
CD = 8, AD = 6, AB = ___ B C A D

111. Examples AB = 12, AD = 4, BC = ___ AC = 7, AB = 12, BD = ___
AD = 4, AB = 16, AC = ___
BD = 6, AB = 8, BC = ___
CD = 8, BD = 16, AD = ___
AD = 3, BD = 24, AC = ___
CD = 8, BD = 16, AD = ___, AC = ___
BC = ___ D

112. 7.5 Proportions and Similar Triangles

113. Side-Splitter Theorem (Triangle Proportionality Theorem) If a line parallel to one side of a triangle intersects the other two sides, then it divides the two side proportionally.

114. Converse of the triangle proportionality theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

115. Triangle Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half as long.

116. Problem

118. Theorem 8.6
If three parallel lines intersect two transversals, then they divide the transversals proportionally.

119. Example 4
Find the length of AB.

120. Triangle-Angle-BisectorTheorem
If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

121. 30 21 24

122. 37.5 9
7.5 x
13.5 y