1. Copyright © Ed2Net Learning, Inc. 1 Algebra I Applications of Proportions

2. Copyright © Ed2Net Learning, Inc. 2 Warm Up 1) 3 = 1 z 8 Solve each proportion. 3) x = 1 3 5 2) f + 3 = 7 12 2 4) -1 = 3 5 2d

3. Copyright © Ed2Net Learning, Inc. 3 Similar Figures Similar figures have the same shape (but not necessarily the same size) and the following properties:  Corresponding sides are proportional. That is, the ratios of the corresponding sides are equal.  Corresponding angles are equal. Corresponding sides of two figures are in the same relative position, and corresponding angles are in the same relative position. Two figures are similar if and only if the lengths of corresponding sides are proportional and all pairs of corresponding angles have equal measures.

4. Copyright © Ed2Net Learning, Inc. 4 If two triangles are similar, then:  they are equiangular.  the corresponding sides are in the same ratio.  the angle included between any two sides of one triangle is equal to the angle included between the corresponding sides of the other triangle. The symbol means is similar to.

5. Copyright © Ed2Net Learning, Inc. 5 ABC XYZ 62° 55° 63° A B C 62° 55° 63° X Z Y corresponding angles corresponding sides. 6 7.5 8 10 AB = BC = CA XY YZ ZX m A = m X m B = m Y m C = m Z Make sure corresponding vertices are in the same order. It would be incorrect to write ABC EFD.

6. Copyright © Ed2Net Learning, Inc. 6 Finding Missing Measures in Similar Figures A) Triangle ABC Triangle A’B’C’. Find the value of x. A B C A’ B’C’ 4 mx 6 m 12 m A corresponds to A’, B corresponds to B’, and C corresponds to C’. BC = CA B’C’ C’A’ x = 12 = 2 4 6 Use cross products. x = 2 × 4 x = 8 The length of BC is 8 m.

7. Copyright © Ed2Net Learning, Inc. 7 x 9 12 4 A B CD A’B' C'D' B) Rectangle ABCD is to rectangle A'B'C'D' where A corresponds to A' and so on. If AB = 12, A'B' = 4, AD = 9, find A'D' Rectangle ABCD is to rectangle A'B'C'D' AB = AD A’B’ A’D’ 12 = 9 4 x Use cross products. 3x = 9 3 3 3x = 9 Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. x = 3 The length of A’D’ is 4.

8. Copyright © Ed2Net Learning, Inc. 8 Now you try! Find the value of x in the diagram. 1) ABCDEF is similar to PQRSTU AB C D E F P Q R S T U 4 m 2 m 6 m x 2) ABC is similar to XYZ 10 x A BC X Y Z 12 18

9. Copyright © Ed2Net Learning, Inc. 9 You can solve a proportion involving similar triangles to find a length that is not easily measured. This method of measurement is called indirect measurement. If two objects form right angles with the ground, you can apply indirect measurement using their shadows. Indirect measurement

10. Copyright © Ed2Net Learning, Inc. 10 Measurement Application A) A pole casts a shadow 45 feet long at the same time that a 6-foot-tall man casts a shadow that is 3 feet long. Write and solve a proportion to find the height of the totem pole. Both the man and the pole form right angles with the ground, and their shadows are cast at the same angle. You can form two similar right triangles. 45 ft x 6 ft 3 ft Next page 

11. Copyright © Ed2Net Learning, Inc. 11 man’s height = man’s shadow pole’s height pole’s shadow We know that, 6 = 3 x 45 So, 3x = 270 Use cross products. 3x = 270 3 3 Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. x = 90 The pole is 90 feet tall.

12. Copyright © Ed2Net Learning, Inc. 12 Now you try! 1)A forest ranger who is 150 cm tall casts a shadow 45 cm long. At the same time, a nearby tree casts a shadow 195 cm long. Write and solve a proportion to find the height of the tree. 2) A woman who is 5.5 feet tall casts a shadow 3.5 feet long. At the same time, a telephone pole casts a shadow 28 feet long. Write and solve a proportion to find the height of the telephone pole.

13. Copyright © Ed2Net Learning, Inc. 13 Scale factor Any two corresponding sides in two similar figures have a common ratio called the scale factor. AB C D EF H G The trapezoids ABCD and EFGH shown above are similar. So, AB = AD = CD = BC = k = scale factor. EF EH HG FG  The ratio of areas of two similar figures is the square of the scale factor.  The ratio of volumes of two similar figures is the cube of the scale factor.

14. Copyright © Ed2Net Learning, Inc. 14 Changing Dimensions A) Every dimension of a 2-by-4-inch rectangle is multiplied by 1.5 to form a similar rectangle. How is the ratio of the perimeters related to the ratio of corresponding sides? How is the ratio of the areas related to the ratio of corresponding sides? A B 4 in 2 in 6 in 3 in Rectangle ARectangle B Perimeter = 2l + 2w 2 (2) + 2 (4) = 122 (6) + 2 (3) = 18 Area = lw4 (2) = 86 (3) = 18 Next page 

15. Copyright © Ed2Net Learning, Inc. 15 Sides = 4 = 2 6 3 A B 4 in 2 in 6 in 3 in  The ratio of the perimeters is equal to the ratio of corresponding sides.  The ratio of the areas is the square of the ratio of corresponding sides. Perimeters = 12 = 2 18 3 Areas = 8 = 4 18 9 A scale factor between 0 and 1 reduces a figure. A scale factor greater than 1 enlarges it.

16. Copyright © Ed2Net Learning, Inc. 16 Changing Dimensions B) Every dimension of a cone with radius 4 cm and height 6 cm is multiplied by 0.5 to form a similar cone. How is the ratio of the volumes related to the ratio of corresponding dimensions? Cone ACone B Volume = 1 ×∏r 2 h 3 1 ×∏(4) 2 ×6 = 32∏ 3 1 ×∏(2) 2 ×3 = 4∏ 3 Next page  6 cm 4 cm 3 cm 2cm A B

17. Copyright © Ed2Net Learning, Inc. 17 Radii = 4 = 2 =2 2 1  The ratio of the volumes is thrice the ratio of the corresponding dimensions. Heights = 6 = 2 =2 3 1 Volumes = 32∏ = 8 = 2 3 4∏ 1 6 cm 4 cm 3 cm 2cm

18. Copyright © Ed2Net Learning, Inc. 18 Now you try! 1) A cuboid has width 12 inches, length 9 inches and height 15 inches. Every dimension of the cuboid is multiplied by 1 to form a similar cuboid. 3 How is the ratio of the volume related to the ratio of the corresponding dimensions?

19. Copyright © Ed2Net Learning, Inc. 19 Assessment 1) Parallelogram ABCD parallelogram EFGH. Find the value of x. AB D C E F H G 10 15 9 x 2) Trapezoids ABCD trapezoids EFGH. Find the value of x. AB C D EF H G 812 6 x

20. Copyright © Ed2Net Learning, Inc. 20 3) Sam is 7 feet tall and casts a shadow 3.5 feet long. At the same time, the pole outside his house casts a shadow 15 feet long. Write and solve a proportion to find the height of the pole. 4) A rectangle has width 12 inches and length 3 inches. Every dimension of the rectangle is multiplied by 1 to form a similar rectangle. 3 How is the ratio of the perimeters related to the ratio of the corresponding sides?

21. Copyright © Ed2Net Learning, Inc. 21 5) Trapezoids ABCD trapezoids EFGH. Find the value of x. 6) Trapezoids ABCDE trapezoids FGHIJ. Find the value of x. F 10 x 15 135 AB CD E H G I 5x 7 14 A BC D E F G H J

22. Copyright © Ed2Net Learning, Inc. 22 A BC D E x 4 4 4 10 4 7) Triangle ABC triangle ADE. Find the value of x. 8) A rectangle has length 12 feet and width 8 feet. Every dimension of the rectangle is multiplied by 3 4 to form a similar rectangle. What is the ratio of the areas of the two triangles?

23. Copyright © Ed2Net Learning, Inc. 23 9) A rectangle has length 5 centimeters and width 3 centimeters. A similar rectangle has length 7.50 centimeters. What is the width in centimeters of this rectangle? 10) A beach ball holds 800 cubic inches of air. Another beach ball has a radius that is half that of the larger ball. How much air does the smaller ball hold?

24. Copyright © Ed2Net Learning, Inc. 24 Let’s review Similar Figures Similar figures have the same shape (but not necessarily the same size) and the following properties:  Corresponding sides are proportional. That is, the ratios of the corresponding sides are equal.  Corresponding angles are equal. Corresponding sides of two figures are in the same relative position, and corresponding angles are in the same relative position. Two figures are similar if and only if the lengths of corresponding sides are proportional and all pairs of corresponding angles have equal measures.

25. Copyright © Ed2Net Learning, Inc. 25 review ABC XYZ 62° 55° 63° A B C 62° 55° 63° X Z Y corresponding angles corresponding sides. 6 7.5 8 10 AB = BC = CA XY YZ ZX m A = m X m B = m Y m C = m Z Make sure corresponding vertices are in the same order. It would be incorrect to write ABC EFD.

26. Copyright © Ed2Net Learning, Inc. 26 review Scale factor Any two corresponding sides in two similar figures have a common ratio called the scale factor. AB C D EF H G The trapezoids ABCD and EFGH shown above are similar. So, AB = AD = CD = BC = k = scale factor. EF EH HG FG  The ratio of areas of two similar figures is the square of the scale factor.  The ratio of volumes of two similar figures is the cube of the scale factor.

27. Copyright © Ed2Net Learning, Inc. 27 review Changing Dimensions B) Every dimension of a cone with radius 4 cm and height 6 cm is multiplied by 0.5 to form a similar cone. How is the ratio of the volumes related to the ratio of corresponding dimensions? Cone ACone B Volume = 1 ×∏r 2 h 3 1 ×∏(4) 2 ×6 = 32∏ 3 1 ×∏(2) 2 ×3 = 4∏ 3 Next page  6 cm 4 cm 3 cm 2cm A B

28. Copyright © Ed2Net Learning, Inc. 28 review Radii = 4 = 2 =2 2 1  The ratio of the volumes is thrice the ratio of the corresponding dimensions. Heights = 6 = 2 =2 3 1 Volumes = 32∏ = 8 = 2 3 4∏ 1 6 cm 4 cm 3 cm 2cm

29. Copyright © Ed2Net Learning, Inc. 29 You did a great job today!