In the figure, square RSTV has its vertices on the sides of A, C, and F are three of the vertices of the cube shown in

A 60° 30° a a B C Figure 5.35 A C D B 60° 30° b a 30° 6 0° 60° a 2 a 2 a In a triangle whose angles measure 30°, 60°, and 90°, the hypotenuse has a length equal to twice the length of the shorter leg, and the length of the longer leg is the product of and the length of the shorter leg. 13 THEOREM 5.5.2 왘 30-60-90 Theorem EXAMPLE 3 Study the picture proof of Theorem 5.5.2. See Figure 5.35a. PICTURE PROOF OF THEOREM 5.5.2 GIVEN: with , , , and PROVE: and PROOF: We reflect across to form an equiangular and therefore equilateral . As shown in Figures 5.35b and 5.35c, we have . To find b in Figure 5.35c, we apply the Pythagorean Theorem. So That is, . AC = a 1 3 b = a 13 b = 13 2a 2 b = 23a 2 b 2 = 3a 2 3a 2 = b 2 , 4a 2 = a 2 + b 2 2a 2 = a 2 + b 2 c 2 = a 2 + b 2 AB = 2a 䉭ABD AC 䉭ABC AC = a 1 3 AB = 2a BC = a m ∠C = 90° m ∠B = 60° m ∠A = 30° 䉭ABC 쮿

2 a

A B C 60 ° 30 ° c a b It would be best to memorize the sketch in Figure 5.36. So that you will more eas- ily recall which expression is used for each side, remember that the lengths of the sides follow the same order as the angles opposite them. Thus, Opposite the 30° smallest angle is a length of shortest side. Opposite the 60° middle angle is length of middle side. Opposite the 90° largest angle is 2a length of longest side. ∠ a 1 3 ∠ ∠ EXAMPLE 4 Find the lengths of the missing sides of each triangle in Figure 5.37. Exs. 8–10 60 ° 30 ° a 2 a 3 a Figure 5.36 60 ° 30 ° a ? 5 S T R ? Figure 5.37 60 ° 30 ° b V W U 20 ? ? Z c Y X 60 ° 30 ° ? ? 9 The converse of Theorem 5.5.1 is true and is described in the following theorem. 5.5 Solution a b c XZ = 2 XY = 2 3 13 = 6 13 L 10.39 = 9 13 3 = 313 L 5.20 ZY = XY 13 : 9 = XY 13 : XY = 9 13 = 9 13 13 13 UV = VW 1 3 = 10 13 L 17.32 UW = 2 VW : 20 = 2 VW : VW = 10 ST = RS 1 3 = 5 13 L 8.66 RT = 2 RS = 2 5 = 10 쮿 쮿 EXAMPLE 5 Each side of an equilateral triangle measures 6 in. Find the length of an altitude of the triangle. 60 60 6 6 a Figure 5.38

3 a

a 60

2 a

b Solution The equilateral triangle shown in Figure 5.38a is separated into two 30°-60°-90° triangles by the altitude. In the 30°-60°-90° triangle in Figure 5.38b, the side of the equilateral triangle becomes the hypotenuse, so and . The altitude lies opposite the 60° angle of the 30°-60°-90° triangle, so its length is or 3 13 in. L 5.20 in. a 1 3 a = 3 2a = 6 Exs. 11–13 If the length of the hypotenuse of a right triangle equals the product of and the length of either leg, then the angles of the triangle measure 45°, 45°, and 90°. 12 THEOREM 5.5.3 GIVEN: The right triangle with lengths of sides a, a and See Figure 5.39. PROVE: The triangle is a 45°-45°-90° triangle Proof In Figure 5.39, the length of the hypotenuse is , where a is the length of either leg. In a right triangle, the angles that lie opposite the congruent legs are also congruent. In a right triangle, the acute angles are complementary, so each of the congruent acute angles measures 45°. a 1 2 a 1 2

2 a

45 45 a a Figure 5.39 쮿 The converse of Theorem 5.5.2 is also true and can be proved by the indirect method. Rather than construct the proof, we state and apply this theorem. See Figure 5.41. EXAMPLE 6 In right , . See Figure 5.40. What are the measures of the angles of the triangle? If , what is the length of or ? ST RS RT = 12 12 RS = ST 䉭RST R S T Figure 5.40

30 a

2 a

Figure 5.41 30 c c 1 2 Figure 5.42 Solution The longest side is the hypotenuse , so the right angle is and . Because , the congruent acute angles are R and T and . Because , . RS = ST = 12 RT = 12 12 m ∠R = m∠T = 45° ∠s RS ⬵ ST m ∠S = 90° ∠S RT 쮿 If the length of the hypotenuse of a right triangle is twice the length of one leg of the tri- angle, then the angle of the triangle opposite that leg measures 30°. THEOREM 5.5.4 An equivalent form of this theorem is stated as follows: EXAMPLE 7 In right with right , and see Figure 5.43. What are the measures of the angles of the triangle? Also, what is the length of ? Solution Because is a right angle, and is the hypotenuse. Because , the angle opposite measures 30°. Thus, and . Because lies opposite the 60° angle, . AC = 12.3 13 L 21.3 AC m ∠B = 60° m ∠A = 30° BC BC = 1 2 AB AB m ∠C = 90° ∠C AC BC = 12.3 AB = 24.6 ∠C 䉭ABC If one leg of a right triangle has a length equal to one-half the length of the hypotenuse, then the angle of the triangle opposite that leg measures 30° see Figure 5.42. 12.3 24.6 B C A Figure 5.43 쮿

1. For the 45°

- 45° - 90° triangle shown, suppose that . Find: a BC b AB

2. For the 45°

- 45° - 90° triangle shown, suppose that Find: a AC b BC

3. For the 30°

- 60° - 90° triangle shown, suppose that . Find: a YZ b XY

4. For the 30°

- 60° - 90° triangle shown, suppose that . Find: a XZ b YZ In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations to two decimal places.

5. Given: Right with

and Find: YZ and XY XZ = 8 m ∠X = 45° 䉭XYZ XY = 2a XZ = a AB = a 1 2. AC = a A C B 45° 45° Exercises 1, 2 Y 60° 30° X Z Exercises 3, 4 X Y Z Exercises 5–8 Exercises 5.5

12. Given: Right with

and Find: DE and DF

13. Given: Rectangle HJKL with diagonals

and Find: HL , HK, and MK m ∠HKL = 30° JL HK EF = 12 13 m ∠E = 2 m ∠F 䉭DEF

6. Given: Right with

and Find: XZ and YZ

7. Given: Right with

and Find: XZ and YZ

8. Given: Right with

and Find: XZ and YZ

9. Given: Right with

and Find: DF and FE DE = 5 m ∠E = 60° 䉭DEF XY = 12 12 m ∠X = 45° 䉭XYZ XY = 10 12 XZ ⬵ YZ 䉭XYZ XY = 10 XZ ⬵ YZ 䉭XYZ F D E Exercises 9–12 L H 6 3 K J M

6 2

R T S V

10. Given: Right with

and Find: DF and DE

11. Given: Right with

and Find: DE and FE FD = 12 13 m ∠E = 60° 䉭DEF FE = 12 m ∠F = 30° 䉭DEF

14. Given: Right with

and Find: RS and ST m ∠STV = 150° RT = 6 12 䉭RST In Exercises 15–19, create drawings as needed.

15. Given: with and

Find: AC and AB

16. Given: Right with

and Find: PM and PN

17. Given: with , ,

and Find: RS and RT

18. Given: with

with W on Find: ZW

19. Given: Square ABCD with diagonals

and intersecting at E Find: DB

20. Given: with angles

as shown in the drawing Find: NM , MP, MQ, PQ, and NQ

21. Given: with angles

as shown in the drawing Find: XY HINT: Compare this drawing to the one for Exercise 20. 䉭XYZ MP ⬜ NQ 䉭NQM DC = 5 13 AC DB YZ = 6 XY ZW ⬜ XY XY ⬵ XZ ⬵ YZ 䉭XYZ ST = 12 m ∠S = 60° m ∠T = 30° 䉭RST MN = 10 12 MP = PN 䉭MNP BC = 6 m ∠A = m∠B = 45° 䉭ABC 60° 30° P 45° 45° 3 N Q M 60° 75° X 45° Y Z 12

22. Given: Rhombus ABCD in which diagonals

and intersect at point E; Find: AC DB = AB = 8 DB AC 8 8 4 4 D 8 4 D 200 ? 400 W E T G S N 30°

23. A carpenter is working with a board that

24. To unload groceries from a delivery truck at the Piggly

Wiggly Market, an 8-ft ramp that rises 4 ft to the door of the trailer is used. What is the measure of the indicated angle ? ∠D

25. A jogger runs along two sides of an open rectangular lot.

If the first side of the lot is 200 ft long and the diagonal distance across the lot is 400 ft, what is the measure of the angle formed by the 200-ft and 400-ft dimensions? To the nearest foot, how much farther does the jogger run by traveling the two sides of the block rather than the diagonal distance across the lot?

26. Mara’s boat leaves the dock at the same time that Meg’s

boat leaves the dock. Mara’s boat travels due east at 12 mph. Meg’s boat travels at 24 mph in the direction N 30° E. To the nearest tenth of a mile, how far apart will the boats be in half an hour? In Exercises 27 to 33, give both exact solutions and approximate solutions to two decimal places.

27. Given: In ,

bisects and Find: DC and DB

28. Given: In ,

bisects and Find: DC and DB

29. Given: is equiangular and

bisects bisects Find: NQ ∠MQN QR ∠MNQ NR NR = 6 䉭MNQ AC = 10 AB = 20 ∠BAC AD 䉭ABC AB = 12 m ∠B = 30° ∠BAC AD 䉭ABC C A B D Exercises 27, 28 R M 6 N Q

30. Given: is an isosceles right triangle

M and N are midpoints of and Find: MN SV ST 䉭STV N 20 V T M S B C D A Exercises 31, 32

31. Given: Right with

and ; point D on ; bisects and Find: BD AB = 12 ∠BAC AD BC m ∠BAC = 60° m ∠C = 90° 䉭ABC

32. Given: Right with

and ; point D on ; bisects and Find: BD AC = 2 13 ∠BAC AD BC m ∠BAC = 60° m ∠C = 90° 䉭ABC is wide. After marking off a point down the side of length , the carpenter makes a cut along with a saw. What is the measure of the angle that is formed? ∠ACB BC 3

3 4

in. 3

3 4

in. C A B 3

3 4

3

3 4

33. Given: with , ,

and Find: AB HINT: Use altitude from C to as an auxiliary line. AB CD BC = 12 m ∠B = 30° m ∠A = 45° 䉭ABC A B C 30° 45° M N T Q P 12 120°

34. Given:

Isosceles trapezoid MNPQ with and ; the bisectors of MQP and NPQ meet at point T on Find: The perimeter of MNPQ MN ∠s m ∠M = 120° QP = 12

35. In regular hexagon ABCDEF,

. Find the exact length of: a Diagonal b Diagonal

36. In regular hexagon ABCDEF, the

length of AB is x centimeters. In terms of x, find the length of: a Diagonal b Diagonal CF BF CF BF AB = 6 inches F E B A D C Exercises 35, 36 Y V Z X W D E C A B

37. In right triangle XYZ, and . Where

V is the midpoint of and , find VW. HINT: Draw XV. m ∠VWZ = 90° YZ YZ = 4 XY = 3

38. Diagonal separates pentagon

ABCDE into square ABCE and isosceles triangle DEC. If and , find the length of diagonal . HINT: Draw DF ⬜ AB. DB DC = 5 AB = 8 EC Segments Divided Proportionally The Angle-Bisector Theorem Ceva’s Theorem Segments Divided Proportionally 5.6 KEY CONCEPTS In this section, we begin with an informal description of the phrase divided proportion- ally . Suppose that three children have been provided with a joint savings account by their parents. Equal monthly deposits have been made to the account for each child since birth. If the ages of the children are 2, 4, and 6 assume exactness of ages for sim- plicity and the total in the account is 7200, then the amount that each child should re- ceive can be found by solving the equation Solving this equation leads to the solution 1200 for the 2-year-old, 2400 for the 4-year- old, and 3600 for the 6-year-old. We say that the amount has been divided proportion- ally. Expressed as a proportion, this is 1200 2 = 2400 4 = 3600 6 2x + 4x + 6x = 7200