Chapter 8

Chapter 8Chapter 8

SimilaritySimilarity

8.18.1

Ratio and ProportionRatio and Proportion

RatiosRatios

Ratio- Comparison of 2 quantities Ratio- Comparison of 2 quantities in the same unitsin the same units

The ratio of a to b can be written The ratio of a to b can be written asas a/ba/b a : ba : b

The denominator cannot be zeroThe denominator cannot be zero

Simplifying RatiosSimplifying Ratios

Ratios should be expressed in simplified Ratios should be expressed in simplified formform 6:8 = 3:46:8 = 3:4

Before reducing, make sure that the Before reducing, make sure that the units are the same.units are the same. 12in : 3 ft12in : 3 ft

12in : 36 in12in : 36 in

1: 31: 3

Examples (page 461)Examples (page 461)

Simplify each ratioSimplify each ratio10.10. 16 students16 students

24 students24 students

12.12. 22 feet22 feet

52 feet52 feet

18.18. 60 cm60 cm

1 m1 m


Simplify each ratioSimplify each ratio20. 20. 2 mi2 mi

3000 ft3000 ft

24. 24. 20 oz.20 oz.

4 lb4 lb

There are 5280 ft in 1 mi.There are 5280 ft in 1 mi.

There are 16 oz in 1 lb.There are 16 oz in 1 lb.


Find the width to length ratioFind the width to length ratio14.14.

16.16.

20 mm

16 mm

2 ft

12 in.

Using Ratios Example 1Using Ratios Example 1

The perimeter of the isosceles The perimeter of the isosceles triangle shown is 56 in. The ratio triangle shown is 56 in. The ratio of LM : MN is 5:4. Find the length of LM : MN is 5:4. Find the length of the sides and the base of the of the sides and the base of the triangle.triangle.

N

L

M


The measures of the angles in a The measures of the angles in a triangle are in the extended ratio triangle are in the extended ratio 3:4:8. Find the measures of the 3:4:8. Find the measures of the angles angles

3x

4x

8x


The ratios of the side lengths of The ratios of the side lengths of ΔΔQRS to the corresponding side QRS to the corresponding side lengths of lengths of ΔΔVTU are 3:2. Find the VTU are 3:2. Find the unknown lengths.unknown lengths.

S

T

R

Q

V

U

2 cm

18 cm

ProportionsProportions

ProportionProportion Ratio = RatioRatio = Ratio Fraction = FractionFraction = Fraction

Means and ExtremesMeans and Extremes Extreme: Mean = Mean: ExtremeExtreme: Mean = Mean: Extreme

Extreme Mean

Mean Extreme

Solving ProportionsSolving Proportions

Solving ProportionsSolving Proportions Cross multiplyCross multiply Let the means equal the extremesLet the means equal the extremes

Example: Example: 3 5

20x

,a c

If thenb d

Properties of ProportionsProperties of Proportions

Cross Product PropertyCross Product Property

Reciprocal PropertyReciprocal Property

,a c b d

If thenb d a c

ad bc

Solving Proportions Solving Proportions Example 1Example 1

9 6

14 x


5

4 10

s s


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

1 7/8 in

2.75 in

8.28.2

Problem solving in Problem solving in Geometry with Geometry with


Properties of ProportionsProperties of Proportions

,a c a b

If thenb d c d

,a c a b c d

If thenb d b d

Example 1Example 1

Tell whether the statement is true Tell whether the statement is true or falseor false A.A.

B. B.

15 3,

10 2

s sIf then

t t

3 5 3 5,

x yIf thenx y x y

Example 2Example 2

In the diagramIn the diagram

Find the length of LQ.Find the length of LQ.

MQ LQ

MN LP

P

M6

N

Q

1513

L 5

Geometric MeanGeometric Mean

Geometric MeanGeometric Mean The geometric mean between two The geometric mean between two

numbers a and b is the positive numbers a and b is the positive number x such that number x such that

ex: 8/4 = ex: 8/4 = 4/2 4/2 a x

x b

Example 3Example 3

Find the geometric mean between Find the geometric mean between 4 and 9.4 and 9.

Similar PolygonsSimilar Polygons

Polygons are similar if and only if Polygons are similar if and only if

the corresponding angles are the corresponding angles are congruent congruent

and and the corresponding sides are the corresponding sides are

proportionate.proportionate.

Similar figures are Similar figures are dilations dilations of each of each other. (They are reduced or other. (They are reduced or enlarged by a scale factor.)enlarged by a scale factor.)

The symbol for similar is The symbol for similar is

Example 1Example 1

Determine if the sides of the polygon are proportionate.

12 m 8 m

8 m

6 m6 m

Example 2

Determine if the sides of the polygon are proportionate.

15 m5 m

9 m

12 m

3 m

4 m

Example 3Example 3Find the missing measurements.

HAPIE NWYRS

HA

P

IE

6

5 4N

W

Y

RS

18 24

21AP =

EI =

SN =

YR =

Example 4Example 4Find the missing measurements.

QUAD SIML

D A

UQ 20

25125º

QD =

MI =

mD =

mU =

mA =

M

I

SL8

95º65º

12

8.4/8.58.4/8.5

Similar TrianglesSimilar Triangles

Similar TrianglesSimilar Triangles

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

Similarity ShortcutsSimilarity Shortcuts

AA Similarity ShortcutAA Similarity Shortcut

If two angles in one triangle are If two angles in one triangle are congruentcongruent to two angles in to two angles in another triangle, then the triangles another triangle, then the triangles are similar.are similar.


SSS Similarity ShortcutSSS Similarity Shortcut

If three sides in one triangle are If three sides in one triangle are proportionalproportional to the three sides in to the three sides in another triangle, then the triangles another triangle, then the triangles are similar.are similar.


SAS Similarity ShortcutSAS Similarity Shortcut

If two sides of one triangle are If two sides of one triangle are proportionalproportional to two sides of to two sides of another triangle and another triangle and

their included angles are their included angles are congruentcongruent, then the triangles are , then the triangles are similar. similar.


We have three shortcuts:We have three shortcuts:

AAAA

SASSAS

SSSSSS

Example 1Example 1

4g

7

69

10.5

Example 2Example 2

h

32

24

50

k

30

Example 3Example 3

42m36

24

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

4m

6m

24m

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

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

10m

6m

4m

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

1.22 7.32

1.82

8.68.6

Proportions and Similar Proportions and Similar TrianglesTriangles


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

TheoremTheorem

Triangle Proportionality TheoremTriangle Proportionality Theorem If a line parallel to one side of a If a line parallel to one side of a

triangle intersects the other two triangle intersects the other two sides, then it divides the two sides sides, then it divides the two sides proportionally.proportionally.Q

S

T

R

U

|| , .RT RU

If TU QS thenTQ US

TheoremTheorem

Converse of the Triangle Converse of the Triangle Proportionality TheoremProportionality Theorem If a line divides two sides of a triangle If a line divides two sides of a triangle

proportionally, then it is parallel to proportionally, then it is parallel to the third side.the third side.Q

S

T

R

U

, || .RT RU

If thenTU QSTQ US

Example 1Example 1

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

U

Y

V

W

X

Example 2Example 2

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

P

9

T

S

26

Q

9.75

24

R

TheoremTheorem

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

TheoremTheorem

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

Example 3Example 3

In the diagram, In the diagram, 1 1 2 2 3, AB 3, AB =6, BC=9, EF=8. What is x?=6, BC=9, EF=8. What is x?

A

ED

x

9

B6

F8

3

2

C

1

Example 4Example 4

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

3

K

ML

15

17

N

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

1.8212.206.10

8.78.7

DilationsDilations

C

PP’

3

9

DilationsDilations

Dilation: Transformation that maps all Dilation: Transformation that maps all points so that the proportion points so that the proportion stands true. stands true.

Enlargement: A dilation which makes Enlargement: A dilation which makes the transformed image larger than the the transformed image larger than the original imageoriginal image

Reduction: A dilation which makes the Reduction: A dilation which makes the transformed image smaller than the transformed image smaller than the original image. original image.

'CP

CP

EnlargementEnlargement

CP

P’3

9

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:

'CP

CP

ReductionReduction

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

'CP

CP

C

PP’

14

6

Find k

Dilations in a coordinate Dilations in a coordinate planeplane

If the center of the dilation is the origin, If the center of the dilation is the origin, the image can be found by multiplying the image can be found by multiplying each coordinate by the scale factoreach coordinate by the scale factor

Example:Example:

Original coordinates: Original coordinates:

(3, 6), (6, 12) and (9, 3)(3, 6), (6, 12) and (9, 3)

Scale factor: 1/3 Scale factor: 1/3

Find the image coordinates. Find the image coordinates.

Chapter 8

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