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Name _________________________________________ Period _______

GEOMETRY NOTES – Chapter 8 Similarity Section 8.1 Ratio and Proportion

GOAL 1: Computing Ratios

If a and b are two quantities that are measured in the same units, then the ratio of a to b is .b

a The ratio of a to

b can also be written as a : b. Ratios should be expressed in simplified form (reduce the ratio to lowest terms:

Example 6:8 reduces to 3:4). If the units are different, then they must be simplified by converting to the same

units first, then reducing the values. Part of the simplification is to remove the unit of measure from the ratio.

Ex. 1 What are the conversions of:

a. 1 foot = _____ inches b. 1 yard = ____ feet c. 1 mile = ___________ feet

d. 1 pint = ______ cups e. 1 quart = _______ cups f. 1 gallon = ________ quarts

g. 1 lb = _______ oz h. 1 m = _______ cm i. 1 km = _________ m

Ex. 2 Simplify the ratio. You may first have to rewrite the fraction so that the numerator and denominator have

the same units.

a. 6 yards

12 yards b.

20 ft

3 yd c.

3 lb

12 oz d.

1.5 m

80 cm

Ex. 3 Find the width to length ratio of each rectangle. Then simplify the ratio.

a. b. c.

Ex. 4 Use the number line to find the ratio of the distances.

a. AB

BD b.

BC

DE c.

CF

AB

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GOAL 2: Using Proportions

______________________________________________________________________ is a proportion.

The numbers a and d are the ___________________ and the numbers b and c are the _________________

of the proportion.

Properties of Proportions

1. CROSS PRODUCT PROPERTY . then , If bcadd

c

b

a

2. RECIPROCAL PROPERTY . then , Ifc

d

a

b

d

c

b

a

Ex. 5 Solve the proportion.

a. x

3

10

15 b.

20

30m

120

c. 5

2y 7

3

y d.

6

x

8

x 3

Section 8.2 Problem Solving in Geometry with Proportions

GOAL 1: Using Properties of Proportions Here are two more properties that are useful in geometry.

Additional Properties of Proportions

3. .c

a then , If

d

b

d

c

b

a

4. . then ,b

a If

d

dc

b

ba

d

c

Ex. 1 Complete the sentence.

a. if a

b

3

4 then

b

a b. if

a

b

3

4 then

a

3

c. if a

b

3

4 then

a b

b d. if

a

b

3

4 then

9

4

3

Ex. 2 Decide whether the statement is true or false.

a. If m

n

4

5, then

n

m

4

5. b. If

m

n

3

6, then

3

nm

6.

c. If m

n

2

3, then

m n

n

5

3. d. If

m

n

3

4, then

m n

n

1

4.

The geometric mean of two positive numbers a and b is the positive number x such that b

x

x

a . If you

solve for x, the formula becomes ,bax x is always a positive number. It is important to note the the

Geometric mean does not equal the arithmetic mean between two numbers. The Arithmetic Mean is

used to find an average. The Geometric Mean is used in proportions.

Ex. 3 Find the geometric mean of the two numbers.

a. 4 and 9 b. 4 and 16 c. 4 and 8

GOAL 2: Using Proportions in Real Life There is more than one correct way to sep up a proportion problem from a word problem.

Ex. 4 The maximum slope of a wheelchair ramp is 1:12. If a wheelchair ramp has a run of 18 feet, what is the

maximum rise?

Ex. 5 Use the diagram and the given information to find the unknown length.

a. Given: AB

BDAC

CE, find BD. b. Given:

MN

NOMP

PQ, find PQ. c. Given:

MN

NOMP

PQ, find PQ.

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Section 8.3 Similar Polygons GOAL 1: Identifying Similar Polygons

When there is a correspondence between two polygons such that their

corresponding angles are _______________and the lengths of corresponding

sides are __________________________ the two polygons are called

____________________________________________.

We name similar figures by their corresponding vertices, and we use a special symbol ~ that means “similar to”.

Ex. 1 List all pairs of congruent angles and write the statement of proportionality for the figure.

Ex. 2 Decide whether the polygons are similar. If so, write a similarity statement.

a. b.

GOAL 2: Using Similar Polygons in Real Life

If two polygons are similar, then the ___________________________ of two corresponding sides is called the

________________________________.

Theorem 8.1 If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their

corresponding side lengths.

Ex. 3 In the diagram at the right, polygon RSTU ~ polygon LMNO.

a. Find the scale factor of RSTU to YLMNO

b. Find the scale factor of LMNO to YRSTU

c. Find the length of NO

d. Find the measure of U .

e. Find the perimeter of LMNO

f. Find the ratio of the perimeter of RSTU to the

Perimeter of LMNO .

WXYZ : YDEFG

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Ex. 4 Tell whether the polygon are always, sometimes, or never similar.

a. Two isosceles triangles

b. Two squares

c. A right and an isosceles triangle

Ex. 4 The two polygons are similar. Find the values of x and y.

a. b.

Section 8.4 Similar Triangles GOAL 1: Using Similarity Theorems How much do we need to know before we can say that two triangles are congruent? You will learn about one

postulate in this section and two theorems in the next section that will give us enough to prove that triangles are ~.

Postulate 25 Angle-Angle (AA) Similarity Postulate

If two angles of one triangle are congruent to two angles of another

triangle, then the two triangles are similar.

.~ then , and If XYZJKLYXZKJLXYZJKL

Ex. 1 The triangles shown are similar. List all the pairs of congruent angles and write the statement of

proportionality.

Ex. 2 Use the diagram to complete the following.

a. TIR : ____ b. TI

IR

RT

c. 24

10

d. 24

12

e. x = f. y =

Ex. 3 Determine whether the triangles can be proved similar. If they are similar, write a similarity statement. If

they are not similar, explain why.

a. b. c.

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GOAL 2: Using Similar Triangles in Real Life

Ex. 4 Find the value of the variable.

a. b.

Section 8.5 Proving Triangles are Similar GOAL 1: Using Similarity Theorems You will study two additional ways to prove that two triangles are similar.

Theorem 8.2 Side-Side-Side (SSS) Similarity Theorem

If the corresponding sides of two triangles are proportional,

then the triangles are similar.

.~ then , If MNPABCRP

CA

QR

BC

PQ

AB

Theorem 8.3 Side-Angle-Side (SAS) Similarity Theorem

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.

.~ then , and If MNPXYZMN

XY

PM

ZXMX

Ex. 1 Name a postulate of theorem that can be used to prove that the two triangles are similar. Then, write a

similarity statement.

1. 2. 3.

Ex. 2 Determine which two of the three given triangles are similar. Find the scale factor for the pair.

4.

5.

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Ex. 3 Are the triangles similar? If so, state the similarity and the postulate or theorem that justifies your answer.

6. 7.

Ex. 4 To determine the height of a very tall pine tree, you place a mirror on the ground and stand where you can

see the top of the tree, s shown. How tall is the tree?

Section 8.6 Proportions and Similar Triangles GOAL 1: Using Proportionality You will learn about four proportionality theorems.

Ex. 1 Use the figure to complete the proportions.

1. AB

AFBC

2. BD

DFEG

3. AD

BDAE

4. AC

AGAB

5. DE

FGAD

6. AB

DFEG

8

x

Ex. 2 Determine whether the statement is true or false.

7. AB

BDAC

CE 8.

AC

CEBC

DE

9. EC

CAED

CB 10.

DB

BAEC

CA

Ex. 3 Find the value of the variable.

11. 12. 13.

Ex. 4 Find the value of the variable.

14. 15. 16.

Ex. 5 Determine whether the given information implies BC DE . Explain.

17. 18. 19.

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Section 8.7 Dilations

GOAL 1: Identifying Dilations A nonrigid transformation in which the image and preimage of a figure are similar is called a dilation.

A dilation is a reduction if its scale factor is less than 1 but greater then 0. 0 < k < 1

A dilation is a enlargement if its scale factor is greater than 1. k > 1

Ex. 1 Identify the dilation and find its scale factor.

1. 2. 3.