5-5 Similar Figures Warm Up Problem of the Day Lesson Presentation Course 3 Warm Up Problem of the Day Lesson Presentation

Warm Up Solve each proportion. = = 1. b = 10 2. y = 8 = = 3. 4. m = 52 30 3 9 = 56 35 y 5 = 1. b = 10 2. y = 8 4 12 p 9 = 56 m 28 26 = 3. 4. m = 52 p = 3

Problem of the Day A rectangle that is 10 in. wide and 8 in. long is the same shape as one that is 8 in. wide and x in. long. What is the length of the smaller rectangle? 6.4 in.

Learn to determine whether figures are similar, to use scale factors, and to find missing dimensions in similar figures.

Vocabulary similar congruent angles scale factor

Similar figures have the same shape, but not necessarily the same size Similar figures have the same shape, but not necessarily the same size. Two triangles are similar if the ratios of the lengths of corresponding sides are equivalent and the corresponding angles have equal measures. Angles that have equal measures are called congruent angles. 43° 43°

Additional Example 1: Identifying Similar Figures Which rectangles are similar? Since the three figures are all rectangles, all the angles are right angles. So the corresponding angles are congruent.

Additional Example 1 Continued Compare the ratios of corresponding sides to see if they are equal. length of rectangle J length of rectangle K width of rectangle J width of rectangle K 10 5 4 2 ? = 20 = 20 The ratios are equal. Rectangle J is similar to rectangle K. The notation J ~ K shows similarity. length of rectangle J length of rectangle L width of rectangle J width of rectangle L 10 12 4 5 ? = 50  48 The ratios are not equal. Rectangle J is not similar to rectangle L. Therefore, rectangle K is not similar to rectangle L.

Which rectangles are similar? Check It Out: Example 1 Which rectangles are similar? 8 ft A 6 ft B C 5 ft 4 ft 3 ft 2 ft Since the three figures are all rectangles, all the angles are right angles. So the corresponding angles are congruent.

Check It Out: Example 1 Continued Compare the ratios of corresponding sides to see if they are equal. length of rectangle A length of rectangle B width of rectangle A width of rectangle B 8 6 4 3 ? = 24 = 24 The ratios are equal. Rectangle A is similar to rectangle B. The notation A ~ B shows similarity. length of rectangle A length of rectangle C width of rectangle A width of rectangle C 8 5 4 2 ? = 16  20 The ratios are not equal. Rectangle A is not similar to rectangle C. Therefore, rectangle B is not similar to rectangle C.

The ratio formed by the corresponding sides is the scale factor.

Additional Example 2A: Using Scale Factors to Find Missing Dimensions A picture 10 in. tall and 14 in. wide is to be scaled to 1.5 in. tall to be displayed on a Web page. How wide should the picture be on the Web page for the two pictures to be similar? Divide the height of the scaled picture by the corresponding height of the original picture. 0.15 1.5 10 = Multiply the width of the original picture by the scale factor. 14 • 0.15 2.1 Simplify. The picture should be 2.1 in. wide.

Additional Example 2B: Using Scale Factors to Find Missing Dimensions A toy replica of a tree is 9 inches tall. What is the length of the tree branch, if a 6 ft tall tree has a branch that is 60 in. long? Divide the height of the replica tree by the height of the 6 ft tree in inches, to find the scale factor. .125 9 72 = Multiply the length of the original tree branch by the scale factor. 60 • .125 7.5 Simplify. The toy replica tree branch should be 7.5 in long.

Check It Out: Example 2A A painting 40 in. tall and 56 in. wide is to be scaled to 10 in. tall to be displayed on a poster. How wide should the painting be on the poster for the two pictures to be similar? Divide the height of the scaled picture by the corresponding height of the original picture. 0.25 10 40 = Multiply the width of the original picture by the scale factor. 56 • 0.25 14 Simplify. The picture should be 14 in. wide.

Check It Out: Example 2B A toy replica of a tree is 6 inches tall. What is the length of the tree branch, if a 8 ft tall tree has a branch that is 72 in. long? Divide the height of the replica tree by the height of the 6 ft tree in inches, to find the scale factor. .0625 6 96 = Multiply the length of the original tree branch by the scale factor. 72 • .0625 4.5 Simplify. The toy replica tree branch should be 4.5 in long.

Additional Example 3: Using Equivalent Ratios to Find Missing Dimensions A T-shirt design includes an isosceles triangle with side lengths 4.5 in, 4.5 in., and 6 in. An advertisement shows an enlarged version of the triangle with two sides that are each 3 ft. long. What is the length of the third side of the triangle in the advertisement? 6 in. x ft 4.5 in. 3 ft = Set up a proportion. 4.5 in. • x ft = 3 ft • 6 in. Find the cross products. 4.5 in. • x ft = 3 ft • 6 in. Divide out the units.

Helpful Hint Draw a diagram to help you visualize the problems

Additional Example 3 Continued Cancel the units. 4.5x = 18 Multiply. x = = 4 18 4.5 Solve for x. The third side of the triangle is 4 ft long.

Check It Out: Example 3 A flag in the shape of an isosceles triangle with side lengths 18 ft, 18 ft, and 24 ft is hanging on a pole outside a campground. A camp t-shirt shows a smaller version of the triangle with two sides that are each 4 in. long. What is the length of the third side of the triangle on the t-shirt? 24 ft x in. 18 ft 4 in. = Set up a proportion. 18 ft • x in. = 24 ft • 4 in. Find the cross products. 18 ft • x in. = 24 ft • 4 in. Divide out the units.

Check It Out: Example 3 Continued Cancel the units. 18x = 96 Multiply. x =  5.3 96 18 Solve for x. The third side of the triangle is about 5.3 in. long.

Lesson Quiz Use the properties of similar figures to answer each question. 1. A rectangular house is 32 ft wide and 68 ft long. On a blueprint, the width is 8 in. Find the length on the blueprint. 17 in. 2. Karen enlarged a 3 in. wide by 5 in. tall photo into a poster. If the poster is 2.25 ft wide, how tall is it? 3.75 ft 3. Which rectangles are similar? A and B are similar.