Similarity

1
Chapter 8

• Similarity

2
8.1

• Ratio and Proportion

3
Ratios

• Ratio- Comparison of 2 quantities in the same
  units
• The ratio of a to b can be written as
• a/b
• a b
• The denominator cannot be zero

4
Simplifying Ratios

• Ratios should be expressed in simplified form
• 68 34
• Before reducing, make sure that the units are the
  same.
• 12in 3 ft
• 12in 36 in
• 1 3

5
Examples (page 461)

• Simplify each ratio
• 10. 16 students
• 24 students
• 12. 22 feet
• 52 feet
• 18. 60 cm
• 1 m

6
Examples (page 461)

• Simplify each ratio
• 20. 2 mi
• 3000 ft
• 24. 20 oz.
• 4 lb
• There are 5280 ft in 1 mi.
• There are 16 oz in 1 lb.

7
Examples (page 461)

• Find the width to length ratio
• 14.
• 16.

8
Using Ratios Example 1

• The perimeter of the isosceles triangle shown is
  56 in. The ratio of LM MN is 54. Find the
  length of the sides and the base of the triangle.

9
Using Ratios Example 2

• The measures of the angles in a triangle are in
  the extended ratio 348. Find the measures of
  the angles

4x
8x
3x
10
Using Ratios Example 3

• The ratios of the side lengths of ?QRS to the
  corresponding side lengths of ?VTU are 32. Find
  the unknown lengths.

2 cm
18 cm
11
Proportions

• Proportion
• Ratio Ratio
• Fraction Fraction
• Means and Extremes
• Extreme Mean Mean Extreme

12
Solving Proportions

• Solving Proportions
• Cross multiply
• Let the means equal the extremes
• Example

13
Properties of Proportions

• Cross Product Property
• Reciprocal Property

14
Solving Proportions Example 1
15
Solving Proportions Example 2
16
Solving Proportions Example 3

• A photo of a building has the measurements shown.
  The actual building is 26 ¼ ft wide. How tall
  is it?

2.75 in
1 7/8 in
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8.2

• Problem solving in Geometry with Proportions

18
Properties of Proportions
19
Example 1

• Tell whether the statement is true or false
• A.
• B.

20
Example 2

• In the diagram
• Find the length of LQ.

21
Geometric Mean

• Geometric Mean
• The geometric mean between two numbers a and b is
  the positive number x such that
• ex 8/4 4/2

22
Example 3

• Find the geometric mean between 4 and 9.

23
Similar Polygons

• Polygons are similar if and only if
• the corresponding angles are congruent
• and
• the corresponding sides are proportionate.

24

• Similar figures are dilations of each other.
  (They are reduced or enlarged by a scale factor.)
• The symbol for similar is ?

25
Example 1
Determine if the sides of the polygon are
proportionate.
26
Example 2
Determine if the sides of the polygon are
proportionate.
27
Example 3
Find the missing measurements. HAPIE ? NWYRS
AP EI SN YR
28
Example 4
Find the missing measurements. QUAD ? SIML
QD MI m?D
m?U m?A
29
8.4/8.5

• Similar Triangles

30
Similar Triangles

•  To be similar, corresponding sides must be
  proportional and corresponding angles are
  congruent.
•  

31
Similarity Shortcuts

• AA Similarity Shortcut
• If two angles in one triangle are congruent to
  two angles in another triangle, then the
  triangles are similar.

32
Similarity Shortcuts

• SSS Similarity Shortcut
• If three sides in one triangle are proportional
  to the three sides in another triangle, then the
  triangles are similar.

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Similarity Shortcuts

• SAS Similarity Shortcut
• If two sides of one triangle are proportional to
  two sides of another triangle and
• their included angles are congruent, then the
  triangles are similar.

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Similarity Shortcuts

• We have three shortcuts
• AA
• SAS
• SSS

35
Example 1
36
Example 2
37
Example 3
38

• 4. A flagpole 4 meters tall casts a 6 meter
  shadow. At the same time of day, a nearby
  building casts a 24 meter shadow. How tall is
  the building?

39

• 5. Five foot tall Melody casts an 84 inch
  shadow. How tall is her friend if, at the same
  time of day, his shadow is 1 foot shorter than
  hers?

40

• 6. A 10 meter rope from the top of a flagpole
  reaches to the end of the flagpoles 6 meter
  shadow. How tall is the nearby football goalpost
  if, at the same moment, it has a shadow of 4
  meters?

41

• 7. Private eye Samantha Diamond places a mirror
  on the ground between herself and an apartment
  building and stands so that when she looks into
  the mirror, she sees into a window. The mirror
  is 1.22 meters from her feet and 7.32 meters from
  the base of the building. Sams eye is 1.82
  meters above the ground. How high is the window?

1.22
7.32
42
8.6

• Proportions and Similar Triangles

43
Proportions

• Using similar triangles missing sides can be
  found by setting up proportions.

44
Theorem

• Triangle Proportionality Theorem
• If a line parallel to one side of a triangle
  intersects the other two sides, then it divides
  the two sides proportionally.

45
Theorem

• Converse of the Triangle Proportionality Theorem
• If a line divides two sides of a triangle
  proportionally, then it is parallel to the third
  side.

46
Example 1

• In the diagram, segment UY is parallel to segment
  VX, UV 3, UW 18 and XW 16. What is the
  length of segment YX?

47
Example 2

• Given the diagram, determine whether segment PQ
  is parallel to segment TR.

48
Theorem

• If three parallel lines intersect two
  transversals, then they divide the transversals
  proportionally.

49
Theorem

• 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.

50
Example 3

• In the diagram, ?1 ? ?2 ? ?3, AB 6, BC9, EF8.
  What is x?

51
Example 4

• In the diagram, ?LKM ? ?MKN. Use the given side
  lengths to find the length of segment MN.

52

• 5. Juanita, who is 1.82 meters tall, wants to
  find the height of a tree in her backyard. From
  the trees base, she walks 12.20 meters along the
  trees shadow to a position where the end of her
  shadow exactly overlaps the end of the trees
  shadow. She is now 6.10 meters from the end of
  the shadows. How tall is the tree?

53
8.7

• Dilations

54
Dilations

• Dilation Transformation that maps all points so
  that the proportion stands true.
• Enlargement A dilation which makes the
  transformed image larger than the original image
• Reduction A dilation which makes the transformed
  image smaller than the original image.

55
Enlargement
An enlargement has a scale factor of k which if
found by the proportion . In an
enlargement k is always greater than 1.
Find k
56
Reduction

• A reduction has a scale factor of k which is
  found by the proportion . In a reduction, 0 lt
  k lt 1.

C
6
P
P
14
Find k
57
Dilations in a coordinate plane

• If the center of the dilation is the origin, the
  image can be found by multiplying each coordinate
  by the scale factor
• Example
• Original coordinates
• (3, 6), (6, 12) and (9, 3)
• Scale factor 1/3
• Find the image coordinates.