Honors Geometry Unit 5 – Similarity and Dilations

Lesson OneProportion and Similarity

Objectives

• I can define similar figures, proportion, ratio

• I can find a scale factor between similar figures

What is a Ratio?

• We discussed ratios in Unit 1

• To compare two quantities: a and b– We write a : b

– Which implies

• Ratios do not include units of measurement

b

a

Ratios• Ratios can also be expressed as decimals

• In this case, the ratio is referred to as a unit ratio

• Ex: Batting Averagehits vs. at bats: 327.0

1

327.0

521

181

Write and Simplify Ratios

SCHOOL The total number of students who participate in sports programs at Central High School is 520. The total number of students in the school is 1850. Find the athlete-to-student ratio to the nearest tenth.

To find this ratio, divide the number of athletes by the total number of students.

Answer: The athlete-to-student ratio is 0.3.

0.3 can be written as

Proportion

• When two ratios are set equal to each other, the equation is called a proportion

• We solve these equations by cross multiplying

Use Cross Products to Solve Proportions

Answer: x = –2

Original proportion

Cross Products

Simplify.

Add 30 to each side.

Divide each side by 24.

A. AB. BC. CD. D

A. n = 9

B. n = 8.9

C. n = 3

D. n = 1.8

Use Proportions to Make Predictions

PETS Monique randomly surveyed 30 students from her class and found that 18 had a dog or a cat for a pet. If there are 870 students in Monique’s school, predict the total number of students with a dog or a cat.

Write and solve a proportion that compares the number of students who have a pet to the number of students in the school.

18 ( 870) = 30x Cross Products Property15,660 = 30x Simplify.

522 = x Divide each side by 30.

Answer: Based on Monique's survey, about 522 students at her school have a dog or a cat for a pet.

Why?

• Multiple figures that have the same shape but are different sizes are known as similar figures

• Similar figures have corresponding angles that are congruent

• Similar figures have corresponding side lengths that are proportional

Similar - Symbol

• To show that two figures are similar, we use the symbol “~”

– We will write similarity statements

– Use this symbol just as you would “=“ or “ “

ExampleSimilar Polygons

The ratio is the same for all 4 sets of corresponding

sides

Use a Similarity Statement

If ΔABC ~ ΔRST, list all pairs of congruent angles and write a proportion that relates the corresponding sides.

ΔABC ~ ΔRST

Congruent Angles: A R, B S, C T

Use Similar Figures to Find Missing Measures

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

Use the congruent angles to write the corresponding vertices in order.polygon ABCDE ~ polygon RSTUV

Answer: x = __92

y = __313

A. AB. BC. CD. D

A. a = 1.4

B. a = 3.75

C. a = 2.4

D. a = 2

The two polygons are similar. Solve for a and b

b = 1.2

b = 2.1

b = 7.2

b = 9.3

Scale Factor• When two figures are similar, the ratio that is found

between all sets of side lengths is called the scale factor– Typically represented with the letter “k”

• Depends on the order of comparison

• If 0 < k < 1, then the scale factor causes the figure to shrink, or reduce in size

• If k > 1, then the scale factor causes the figure to grow in size, or enlarge

• What happens if k = 1?

Scale Factor

Use a Scale Factor to Find Perimeter

If ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV and the perimeter of each polygon.

Use a Scale Factor to Find Perimeter

The scale factor ABCDE to RSTUV is or . ___AEVU

__47

Write a proportion to find the length of DC.

Since DC AB and AE DE, the perimeter of ABCDE is 6 + 6 + 6 + 4 + 4 or 26.

Write a proportion.

4(10.5)= 7 ● DC Cross Products Property6 = DC Divide each side by 7.

Use a Scale Factor to Find Perimeter

Use the perimeter of ABCDE and scale factor to write a proportion. Let x represent the perimeter of RSTUV.

Theorem 7.1

Substitution

4x = (26)(7) Cross Products Property

x = 45.5 Solve.

A. AB. BC. CD. D

A. LMNOP = 40, VWXYZ = 30

B. LMNOP = 32, VWXYZ = 24

C. LMNOP = 45, VWXYZ = 40

D. LMNOP = 60, VWXYZ = 45

If LMNOP ~ VWXYZ, find the perimeter of each polygon.