7-6 Similar Figures Warm Up Problem of the Day Lesson Presentation Pre-Algebra

7-6 Similar Figures Warm Up Solve each proportion. = = 1. b = 10 2. Pre-Algebra 7-6 Similar Figures Warm Up Solve each proportion. b 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 plane figure is dilated and gets 50% larger. What scale factor should you use to dilate the figure back to its original size? (Hint: The answer is not ). 1 2 2 3

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

Vocabulary similar

The heights of letters in newspapers and on billboards are measured using points and picas. There are 12 points in 1 pica and 6 picas in one inch. A letter 36 inches tall on a billboard would be 216 picas, or 2592 points. The first letter in this paragraph is 12 points.

Congruent figures have the same size and shape Congruent figures have the same size and shape. Similar figures have the same shape, but not necessarily the same size. The A’s in the table are similar. The have the same shape, but they are not the same size. The ratio formed by the corresponding sides is the scale factor.

Additional Example 1: 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? To find the scale factor, divide the known measurement of the scaled picture by the corresponding measurement of the original picture. 0.15 1.5 10 = 0.15 Then multiply the width of the original picture by the scale factor. 2.1 14 • 0.15 = 2.1 The picture should be 2.1 in. wide.

Try This: Example 1 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? To find the scale factor, divide the known measurement of the scaled painting by the corresponding measurement of the original painting. 0.25 10 40 = 0.25 Then multiply the width of the original painting by the scale factor. 14 56 • 0.25 = 14 The painting should be 14 in. wide.

Additional Example 2: 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? Set up a proportion. 6 in. x ft 4.5 in. 3 ft = 4.5 in. • x ft = 3 ft • 6 in. Find the cross products. 4.5 in. • x ft = 3 ft • 6 in. in. • ft is on both sides

Additional Example 2 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.

Try This: Example 2 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? Set up a proportion. 24 ft x in. 18 ft 4 in. = 18 ft • x in. = 24 ft • 4 in. Find the cross products. 18 ft • x in. = 24 ft • 4 in. in • ft is on both sides

Try This: Example 2 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.

The following are matching, or corresponding: A and X Remember! The following are matching, or corresponding: A and X A X Z Y C B AB and XY BC and YZ B and Y AC and XZ C and Z

Additional Example 3: 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 3 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.

Which rectangles are similar? Try This: Example 3 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.

The ratios are not equal. Rectangle A is not similar to rectangle C. Try This: Example 3 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.

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. 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. Which rectangles are similar?