All pairs of corresponding sides are proportional. DEFINITION

In an extended proportion, the ratios must all be equal to the same constant value. By designating this number which is often called the “constant of proportionality” by k, we see that It follows that , , and . In Example 3, this con- stant of proportionality had the value , which means that the length of each side of the larger triangle was twice the length of the corresponding side of the smaller triangle. If , the similarity leads to an enlargement, or stretch. If , the similarity results in a shrink. The constant of proportionality is also used to scale a map, a diagram, or a blue- print. As a consequence, scaling problems can be solved by using proportions. 0 6 k 6 1 k 7 1 k = 2 DE = k AB FE = k CB DF = k AC DF AC = k, FE CB = k, and DE AB = k Exs. 5–10 EXAMPLE 4 On a map, a length of 1 in. represents a distance of 30 mi. On the map, how far apart should two cities appear if they are actually 140 mi apart along a straight line? Solution Where the map distance desired in inches, Then and x = 4 2 3 in. 30x = 140

1 30

= x 140 x = 쮿 D A E C B 16 x x 3 Figure 5.8 EXAMPLE 5 In Figure 5.8, with . If , , and , find the length BC. Solution From the similar triangles, we have . With and representing the lengths of the congruent segments and by x, we have Substituting into the proportion, we have It follows that Now x or BC equals 4 or 12. Each length is acceptable, but the scaled drawings differ, as illustrated in Figure 5.9 on next page. x - 4x - 12 = 0 x 2 - 16x + 48 = 0 16x - x 2 = 48 x16 - x = 3

16 3

x = 16 - x 16 16 = AE + x so AE = 16 - x BC EC AC = AE + EC DE BC = AE AC EC = BC AC = 16 DE = 3 ∠ADE ⬵ ∠B 䉭ABC 䉭ADE The following example uses a method called shadow reckoning. This method of cal- culating a length dates back more than 2500 years when it was used by the Greek mathe- matician, Thales, to estimate the height of the pyramids in Egypt. In Figure 5.10, the method assumes correctly that . Note that and . ∠C ⬵ ∠F ∠A ⬵ ∠D 䉭ABC 䉭DEF

16 3

4 4 12 a Figure 5.9 16 4 12 b 3 12 쮿 EXAMPLE 6 Darnell is curious about the height of a flagpole that stands in front of his school. Darnell, who is 6 ft tall, casts a shadow that he paces off at 9 ft. He walks the length of the shadow of the flagpole, a distance of 30 ft. How tall is the flagpole? D F E C B h 9 30 6 A Figure 5.10 Solution In Figure 5.10, . From similar triangles, we know that or by interchanging the means. Where h is the height of the flagpole, substitution into the second proportion leads to The height of the flagpole is 20 ft. 6 9 = h 30 : 9h = 180 : h = 20 AC BC = DF EF AC DF = BC EF 䉭ABC 䉭DEF 쮿 Exs. 11–13 Exercises 5.2

1. a What is true of any pair of corresponding angles of two

similar polygons? b What is true of any pairs of corresponding sides of two similar polygons? 2. a Are any two quadrilaterals similar? b Are any two squares similar? 3. a Are any two regular pentagons similar? b Are any two equiangular pentagons similar? 4. a Are any two equilateral hexagons similar? b Are any two regular hexagons similar? In Exercises 5 and 6, refer to the drawing.

5. a Given that ,

, and , write a statement claiming that the triangles shown are similar. b Given that , , and , write a statement claiming that the triangles shown are similar. B 4 T C 4 X A 4 N C 4 N B 4 T A 4 X B C A Exercises 5, 6 T N X A B E C D

6. a If , which angle of

corresponds to of ? b If , which side of corresponds to side of ? 7. A sphere is the three-dimensional surface that contains all points in space lying at a fixed distance from a point known as the center of the sphere. Consider the two spheres shown. Are these two spheres similar? Are any two spheres similar? Explain. 䉭ABC AC 䉭XTN 䉭ABC 䉭XTN 䉭XTN ∠N 䉭ABC 䉭ABC 䉭XTN

8. Given that rectangle ABCE is similar to rectangle MNPR

and that , what can you conclude regarding pentagon ABCDE and pentagon MNPQR? 䉭CDE 䉭PQR M R N P Q R P M S Q N

9. Given: , , ,

, , , Find: a c NP b d QS m ∠P m ∠N MP = 12 RS = 7 QR = 6 MN = 9 m ∠R = 82° m ∠M = 56° 䉭MNP 䉭QRS

10. Given: ,

, , , , Find: a b c AC d CB

11. a Does the similarity relationship have a

reflexive property for triangles and polygons in general? b Is there a symmetric property for the similarity of triangles and polygons? c Is there a transitive property for the similarity of triangles and polygons?

12. Using the names of properties from Exercise 11, identify

the property illustrated by each statement: a If , then . b If , , and , then . c

13. In the drawing, . If

, , and , find FG.

14. In the drawing, . If

, , and , find HJ. FG = 5 KF = 8 HK = 6 䉭HJK 䉭FGK HJ = 4 KF = 8 HK = 6 䉭HJK 䉭FGK 䉭1 䉭1 䉭1 䉭4 䉭3 䉭4 䉭2 䉭3 䉭1 䉭2 䉭2 䉭1 䉭1 䉭2 m ∠RPC m ∠B AB = 26 PR = 13 CR = 12 PC = 5 m ∠A = 67° 䉭ABC 䉭PRC C R B A P F H G K J Exercises 13, 14 a B C A D Exercises 15–20

15. Quadrilateral .

If , , and , find . m ∠K m ∠D = 98° m ∠J = 128° m ∠A = 55° ABCD quadrilateral HJKL

16. Quadrilateral . If

, , , and , find x.

17. Quadrilateral . If

, , , and , find n.

18. Quadrilateral . If

, , , and , find the length of diagonal not shown.

19. Quadrilateral . If

, , and , find .

20. Quadrilateral . If

, and , what types of quadrilaterals are ABCD and HJKL? m ∠B = 110° m ∠A = m∠K = 70° ABCD quadrilateral HJKL m ∠L m ∠D = 3x - 6 m ∠H = 68° m ∠A = 2x + 4 ABCD quadrilateral HJKL HK HL = 12 DC = 6 AD = 8 m ∠D = 90° ABCD quadrilateral HJKL JK = n + 3 HJ = 10 BC = n AB = 5 ABCD quadrilateral HJKL m ∠K = 2x - 45 m ∠D = x + 35 m ∠J = x + 50 m ∠A = x ABCD quadrilateral HJKL b H L J K In Exercises 21 to 24, .

21. Given: , ,

Find: BC

22. Given: , ,

Find: AB HINT: Find DB first.

23. Given: , ,

Find: BC

24. Given: , ,

Find: AE

25. Pentagon ABCDE pentagon GHJKL not shown,

, and . If the perimeter of ABCDE is 50, find the perimeter of GHJKL.

26. Quadrilateral MNPQ quadrilateral WXYZ not shown,

, and . If the longest side of MNPQ is of length 8, find the length of the longest side of WXYZ.

27. A blueprint represents the 72-ft length of a building by a

line segment of length 6 in. What length on the blueprint would be used to represent the height of this 30-ft-tall building?

28. A technical drawing shows the -ft lengths of the legs of

a baby’s swing by line segments 3 in. long. If the diagram should indicate the legs are ft apart at the base, what length represents this distance on the diagram? In Exercises 29 to 32, use the fact that triangles are similar.

29. A person who is walking away from a 10-ft lamppost

casts a shadow 6 ft long. If the person is at a distance of 10 ft from the lamppost at that moment, what is the person’s height? 2 1 2 3 1 2 YZ = 7 PQ = 5 GH = 9 AB = 6 DB = AE AC = 18 AD = 4 EC = BC AC = 20 DE = 4 DB = BC AD = 8 DE = 5 EC = BC AE = 6 DE = 4 䉭ADE 䉭ABC C A B D E Exercises 21–24

30. With 100 ft of string out, a kite is 64 ft above ground

level. When the girl flying the kite pulls in 40 ft of string, the angle formed by the string and the ground does not change. What is the height of the kite above the ground after the 40 ft of string have been taken in?

31. While admiring a rather tall tree, Fred notes that the

shadow of his 6-ft frame has a length of 3 paces. On the level ground, he walks off the complete shadow of the tree in 37 paces. How tall is the tree?

32. As a garage door closes, light is

cast 6 ft beyond the base of the door as shown in the accompanying drawing by a light fixture that is set in the garage ceiling 10 ft back from the door. If the ceiling of the garage is 10 ft above the floor, how far is the garage door above the floor at the time that light is cast 6 ft beyond the door?

33. In the drawing,

with transversals and m. If D and C are the midpoints of and , respectively, then is trapezoid ABCD similar to trapezoid DCFE?

34. In the drawing, .

Suppose that transversals and m are also parallel. D and C are the midpoints of and , respectively. Is parallelogram ABCD similar to parallelogram DCFE?

35. Given , a second triangle

is constructed so that and . a Is congruent to ? b Using intuition appearance, does it seem that is similar to ? 䉭ABC 䉭XTN ∠B ∠T ∠N ⬵ ∠C ∠X ⬵ ∠A 䉭XTN 䉭ABC BF AE Í AB 7 Í DC 7 Í EF BF AE Í AB 7 Í DC 7 Í EF 10 10 6 B C A T N X R S T U W

36. Given

, a second triangle is constructed so that , , and . a What is the constant value of the ratios , , and ? b Using intuition appearance, does it seem that is similar to ? 䉭RST 䉭UVW WU RT VW ST UV RS WU = 2RT VW = 2ST UV = 2RS 䉭UVW 䉭RST A D m E B C F Exercises 33, 34