both sides inverted TEST 534

Discover THE GOLDEN RATIO It is believed that the “ideal” rectangle is determined when a square can be removed in such a way as to leave a smaller rectangle with the same shape as the original rectangle. As we shall find, the rectangles are known as similar in shape. Upon removal of the square, the similarity in the shapes of the rectangles requires that . To discover the relationship between L and W, we choose W 1 and solve the equation for L. The solution is . The ratio comparing length to width is known as the golden ratio. Because when W 1 and , the ideal rectangle has a length that is approximately 1.62 times its width; that is, . L L 1.62W 1 + 15 2 L 1.62 = L = 1 + 15 2 L = 1 + 15 2 1 L = L - 1 1 = W L = L - W W W W L–W L Exercises 5.1 In Exercises 1 to 4, give the ratios in simplified form.

1. a 12 to 15 c 1 ft to 18 in.

b 12 in. to 15 in. d 1 ft to 18 oz

2. a 20 to 36 c 20 oz to 2 lb 1 lb = 16 oz

b 24 oz to 52 oz d 2 lb to 20 oz

3. a 15:24 c 2 m:150 cm 1 m = 100 cm

b 2 ft:2 yd 1 yd = 3 ft d 2 m:1 lb

4. a 24:32 c 150 cm:2 m

b 12 in.:2 yd d 1 gal:24 mi In Exercises 5 to 14, find the value of x in each proportion.

5. a b

6. a b

7. a b

8. a b

9. a

b

10. a b

11. a b

12. a b

13. a b

14. a

b x + 2 x = 2x x - 2 x + 1 x = x x - 1 x + 1 x - 1 = 2x 5 x + 1 x = 2x 3 x + 1 3 = 5 x - 2 x + 1 2 = 7 x - 1 2x + 1 x + 1 = 14 3x - 1 x + 1 x =

10 2x

x - 2 5 = 12 x + 2 x + 1 3 = 10 x + 2 x 6 = 3 x x 4 = 7 x 32 x = x 2 9 x = x 16 x + 1 6 = 4x - 1 18 x - 3 8 = x + 3 24 x + 1 6 = 10 12 x - 1 10 = 3 5 7 x = 21 24 x 4 = 9 12

15. Sarah ran the 300-m hurdles in 47.7 sec. In meters per

second, find the rate at which Sarah ran. Give the answer to the nearest tenth of a meter per second.

16. Fran has been hired to sew the dance troupe’s dresses for

the school musical. If of material is needed for the four dresses, find the rate that describes the amount of material needed for each dress. In Exercises 17 to 22, use proportions to solve each problem.

17. A recipe calls for 4 eggs and 3 cups of milk. To prepare

for a larger number of guests, a cook uses 14 eggs. How many cups of milk are needed?

18. If a school secretary copies 168 worksheets for a class of

28 students, how many worksheets must be prepared for a class of 32 students?

19. An electrician installs 20 electrical outlets in a new six-

room house. Assuming proportionality, how many outlets should be installed in a new construction having seven rooms? Round up to an integer.

20. The secretarial pool 15 secretaries in all on one floor of

a corporate complex has access to four copy machines. If there are 23 secretaries on a different floor, approximately what number of copy machines should be available? Assume a proportionality.

21. Assume that AD is the geometric mean of BD and DC in

shown in the accompanying drawing. a Find AD if and . b Find BD if and . DC = 8 AD = 6 DC = 8 BD = 6 䉭ABC 13 1 3 yd D B C A Exercises 21, 22

22. In the drawing for Exercise 21, assume that AB is the

geometric mean of BD and BC. a Find AB if and . b Find DC if and .

23. The salaries of a secretary, a salesperson, and a vice

president for a retail sales company are in the ratio 2:3:5. If their combined annual salaries amount to 124,500, what is the annual salary of each?

24. If the measures of the angles of a quadrilateral are in the

ratio of 2:3:4:6, find the measure of each angle.

25. The measures of two complementary angles are in the

ratio 4:5. Find the measure of each angle, using the two methods shown in Example 7.

26. The measures of two supplementary angles are in the ratio

of 2:7. Find the measure of each angle, using the two methods of Example 7.

27. If 1 in. equals 2.54 cm, use a proportion to convert 12 in.

to centimeters. HINT:

28. If 1 kg equals 2.2 lb, use a proportion to convert

12 pounds to kilograms.

29. For the quadrilaterals shown, . If

, , and , find YZ. PQ = 6 WX = 3 MN = 7 MN WX = NP XY = PQ YZ = MQ WZ 2.54 cm 1 in. = x cm 12 in. BC = 15 AB = 10 DC = 10 BD = 6

32. Two numbers a and b are in the ratio 2:3. If both numbers

are decreased by 2, the ratio of the resulting numbers becomes 3:5. Find a and b.

33. If the ratio of the measure of the complement of an angle

to the measure of its supplement is 1:3, find the measure of the angle.

34. If the ratio of the measure of the complement of an angle

to the measure of its supplement is 1:4, find the measure of the angle.

35. On a blueprint, a 1-in. scale corresponds to 3 ft. To show a

room with actual dimensions 12 ft wide by 14 ft long, what dimensions should be shown on the blueprint?

36. To find the golden ratio see the Discover activity on page

226, solve the equation for L. HINT: You will need the Quadratic Formula. 1 L = L - 1 1 M N P Q Z Y X W Exercises 29, 30 W W L–W L

30. For this exercise, use the drawing and extended ratio of

Exercise 29. If and , find MQ.

31. Two numbers a and b are in the ratio 3:4. If the first

number is decreased by 2 and the second is decreased by 1, they are in the ratio 2:3. Find a and b. WZ = 3 1 2 NP = 2 XY

37. Find:

a The exact length of an ideal rectangle with width W = 5 by solving b The approximate length of an ideal rectangle with width by using

38. Prove: If

where a, b, c, and d are nonzero, then

39. Prove: If

where and , then a + b b = c + d d . d Z b Z a b = c d a c = b d . a b = c d L L 1.62W W = 5 5 L = L - 5 5 Similar Polygons Congruent Polygons Corresponding Vertices, Angles, and Sides Similar Polygons 5.2 KEY CONCEPTS When two geometric figures have exactly the same shape, they are similar; the symbol for “is similar to” is . When two figures have the same shape and all correspon- ding parts have equal ⫽ measures, the two figures are congruent ⬵. Note that the symbol for congruence combines the symbols for similarity and equality. In fact, we in- clude the following property for emphasis. Two congruent polygons are also similar polygons. Two-dimensional figures such as and in Figure 5.3 can be similar, but it is also possible for three-dimensional figures to be similar. Similar orange juice containers are shown in Figures 5.4 a and 5.4 b. Informally, two figures are “similar” if one is an enlargement of the other. Thus a tuna fish can and an orange juice can are not similar, even if both are right-circular cylinders [see Figures 5.4b and 5.4c]. We will consider cylinders in greater detail in Chapter 9. 䉭DEF 䉭ABC a A C B Figure 5.3 b D F E O.J. 6 ounces O.J. 6 ounces Figure 5.4 O.J. 1 6 oun ces O.J. 1 6 ounces T U N A T U N A Our discussion of similarity will generally be limited to plane figures. For two polygons to be similar, it is necessary that each angle of one polygon be congruent to the corresponding angle of the other. However, the congruence of angles is not sufficient to establish the similarity of polygons. The vertices of the congruent an- gles are corresponding vertices of the similar polygons. If in one polygon is con- gruent to in the second polygon, then vertex A corresponds to vertex M, and this is symbolized ; we can indicate that corresponds to by writing . A pair of angles like and are corresponding angles, and the sides determined by consecutive and corresponding vertices are corresponding sides of the similar polygons. For instance, if and , then corresponds to . MN AB B 4 N A 4 M ∠M ∠A ∠A 4 ∠M ∠M ∠A A 4 M ∠M ∠A Discover When a transparency is projected onto a screen, the image created is similar to the projected figure. EXAMPLE 1 Given similar quadrilaterals ABCD and HJKL with congruent angles as indicated in Figure 5.5, name the vertices, angles, and sides that correspond to each other. a B C A D Figure 5.5 b H L J K Solution Because , it follows that Similarly, D 4 L and ∠D 4 ∠L C 4 K and ∠C 4 ∠K B 4 J and ∠B 4 ∠J A 4 H and ∠A 4 ∠H ∠A ⬵ ∠H a b c