Understanding Similar Figures and Ratios

1. Ch. 7 Learning Goal: Ratios & Proportions • Learn to find equivalent ratios to create proportions (7-1) • Learn to work with rates and ratios (7-2) • Learn to use one or more conversion factors to solve rate problems (7-3) • Learn to solve proportions (7-4) • Learn to identify and create dilations of plane figures (7-5) • Learn to determine whether figures are similar, to use scale factors, and to find missing dimensions similar figures (7-6) • Learn to make comparisons between and find dimensions of scale drawings and actual objects (7-7) • Learn to make comparisons between and find dimensions of scale models and actual objects (7-8) • Learn to make scale models of solid figures (7-9)

2. Page 364 #7-12 & #20-24 (SR)

3. Pre-Algebra Homework Page 370-371 #1-6 & #21-26 (Spiral Review)

4. 7-6 Similar Figures Warm Up Problem of the Day Lesson Presentation Pre-Algebra

5. Today’s Learning Goal Assignment Learn to determine whether figures are similar, to use scale factors, and to find missing dimensions in similar figures.

6. Vocabulary similar

7. 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.

8. 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. They have the same shape, but they are not the same size. The ratio formed by the corresponding sides is the scale factor.

9. 1.5 10 0.15 = 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 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.

10. 10 40 0.25 = 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 Then multiply the width of the original painting by the scale factor. 14 56 • 0.25 = 14 The painting should be 14 in. wide.

11. 4.5 in. 3 ft 6 in. x ft = 4.5 in. • x ft = 3 ft • 6 in. in. • ft is on both sides 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. 4.5 in. • x ft = 3 ft • 6 in. Find the cross products.

12. 18 4.5 x = = 4 Additional Example 2 Continued Cancel the units. 4.5x = 3 • 6 Multiply 4.5x = 18 Solve for x. The third side of the triangle is 4 ft long.

13. 18 ft 4 in. 24 ft x in. = 18 ft • x in. = 24 ft • 4 in. in • ft is on both sides 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. 18 ft • x in. = 24 ft • 4 in. Find the cross products.

14. 96 18 x =  5.3 Try This: Example 2 Continued Cancel the units. 18x = 24 • 4 Multiply 18x = 96 Solve for x. The third side of the triangle is about 5.3 in. long.

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

16. 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.

17. length of rectangle J length of rectangle K 10 5 4 2 width of rectangle J width of rectangle K ? = length of rectangle J length of rectangle L 10 12 4 5 width of rectangle J width of rectangle L ? = Additional Example 3 Continued Compare the ratios of corresponding sides to see if they are equal. 20 = 20 The ratios are equal. Rectangle J is similar to rectangle K. The notation J ~ K shows similarity. 50  48 The ratios are not equal. Rectangle J is not similar to rectangle L.

18. 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.

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

20. 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.