Lesson 44 Applying Similarity

Vocabulary Review from Lesson 41 A ratio is a comparison of two values by division A statement that two ratios are equal is called a proportion In the proportion 𝑎 𝑏 = 𝑐 𝑑 , a and d are the extremes, and b and c are the means Two figures that have the same shape, but not necessarily the same size, are similar In similar polygons, the corresponding angles are congruent and the corresponding sides are proportional A similarity ratio is the ratio of two corresponding linear measurements in a pair of similar figures

Use Similarity to find the unknown Measures Start with a ratio of two known corresponding sides and then write a proportion 14 18 = 𝑥 12 18𝑥=12 14 𝑥= 12(14) 18 𝑥= 28 3 or 9 1 3 14 18 = 13 𝑦 14𝑦=13 18 𝑦= 13(18) 14 𝑦= 117 7 or 16 5 7

Applying Similarity to solve for unknowns The pentagons are regular and similar to each other with a similarity ratio of 4:5 Find x and y Find the perimeter of each Write a similarity ratio of their perimeters

Applying Similarity to solve for unknowns The pentagons are regular and similar to each other with a similarity ratio of 4:5 Find x and y 2𝑦+9=45 2𝑦=36 𝑦=18 4 5 = 𝑥 2 +11 45 5 𝑥 2 +11 =4 45 𝑥 2 +11=36 𝑥 2 =25 𝑥=±5

Applying Similarity to solve for unknowns The pentagons are regular and similar to each other with a similarity ratio of 4:5 Find the perimeter of each 5 5 2 +11 5 25+11 5 36 180 𝑚 & 5 45 225 𝑚

Applying Similarity to solve for unknowns The pentagons are regular and similar to each other with a similarity ratio of 4:5 Write a ratio of their perimeters 180 𝑚 225 𝑚 4 5 What do you notice about the ratio of the perimeters and the similarity ratio?

Theorem 44-1 If two polygons are similar, then the ratio of their perimeters is equal to the ratio of their corresponding sides. If the similarity ratio is a:b, then the ratio of their perimeters is also a:b.

Applying Similarity to solve the perimeter Figures EFGH and IJKL are similar. Their corresponding sides have a ratio of 7:3. If the perimeter of figure EFGH is 35 inches, what is the perimeter of figure IJKL? Do we really need to know the dimensions of the sides to answer this question? 7 3 = 35 𝑃 7𝑃=3 35 𝑃= 3(35) 7 𝑃=15 𝑖𝑛𝑐ℎ𝑒𝑠

Application: Map Scales Sharon plans to run 6000 meters a day in training to prepare for a race. The park where Sharon runs is in the shape of a regular hexagon. The side length of the park measures 2 centimeters on the map with a scale of 1 cm: 100 m. How many times does Sharon need to run along the perimeter of the park to complete her daily training? Start by finding the perimeter of the park on the map 𝑃=6 2 𝑃=12 1 100 = 12 𝑥 𝑥=1200 𝑚 Does this answer the question? If not, what else do we need to do? 6000 𝑚 1200 𝑚 =5

In conclusion… Can we apply Thm. 44-1 to the circumference of a circle? Yes, because circumference is perimeter for circles How would doubling the radius change the circumference? It would double the circumference Applying similarity will prepare you for: Lesson 60: Proportionality Theorems Lesson 66: Finding Perimeters and Areas of Regular Polygons Lesson 87: Area Ratios of Similar Polygons Lesson 99: Volume Ratios of Similar Solids