1. Unit 5 – Similarity and Dilations Lesson One Proportion and Similarity
Honors Geometry
Unit 5 – Similarity and Dilations
Lesson One
Proportion and Similarity

2. Objectives I can define similar figures, proportion, ratio
I can find a scale factor between similar figures

3. 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

4. Ratios Ratios can also be expressed as decimals
In this case, the ratio is referred to as a
unit ratio
Ex: Batting Average
hits vs. at bats:

5. 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 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.
0.3 can be written as
Answer: The athlete-to-student ratio is 0.3.

6. Proportion
When two ratios are set equal to each other, the equation is called a proportion
We solve these equations by cross multiplying

7. Use Cross Products to Solve Proportions
Original proportion
Cross Products
Simplify.
Add 30 to each side.
Divide each side by 24.
Answer: x = –2

8. A. n = 9
B. n = 8.9
C. n = 3
D. n = 1.8 A B C D

9. 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 Property
15,660 = 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.

10. 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

11. 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 “ “

12. Example Similar Polygons
The ratio is the same for all 4 sets of corresponding sides

13. ΔABC ~ ΔRST 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

14. 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 = __ 9 2
y = __ 3 13

15. A B C D The two polygons are similar. Solve for a and b A. a = 1.4
C. a = 2.4
D. a = 2
b = 1.2
b = 2.1
b = 7.2
b = 9.3 A B C D

16. 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?

17. Scale Factor

18. Use a Scale Factor to Find Perimeter
If ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV and the perimeter of each polygon.

19. Use a Scale Factor to Find Perimeter
The scale factor ABCDE to RSTUV is or
___ AE VU __ 4 7
Write a proportion to find the length of DC.
Write a proportion.
4(10.5) = 7 ● DC Cross Products Property
6 = DC Divide each side by 7.
Since DC  AB and AE  DE, the perimeter of ABCDE is or 26.

20. 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.

21. A B C D If LMNOP ~ VWXYZ, find the perimeter of each polygon.
A. LMNOP = 40, VWXYZ = 30
B. LMNOP = 32, VWXYZ = 24
C. LMNOP = 45, VWXYZ = 40
D. LMNOP = 60, VWXYZ = 45 A B C D