582 CHAPTER 8 Applications of Trigonometric Functions 54. Another Cow Problem If the barn in Problem 53 is rectangular,10 feet by 20 feet, what is the maximum grazing area for the cow? 55. Perfect Triangles A perfect triangle is one having integers for sides for which the area is numerically equal to the perimeter. Show that the triangles with the given side lengths are perfect. (a) 9, 10, 17 (b) 6, 25, 29 Source: M.V. Bonsangue, G. E. Gannon, E. Buchman, and N. Gross,“In Search of Perfect Triangles,” Mathematics Teacher, Vol. 92, No. 1, 1999: 56–61. 56. If h h , , 1 2 and h3 are the altitudes dropped from P, Q, and R, respectively, in a triangle (see the figure), show that + + = h h h s K 1 1 1 1 2 3 where K is the area of the triangle and ( ) = + + s a b c 1 2 . [Hint: = h K a 2 . 1 ] h1 a Q P R B C A c b 57. Show that a formula for the altitude h from a vertex to the opposite side a of a triangle is = h a B C A sin sin sin 58. Challenge Problem A triangle has vertices ( ) A 0, 0 , ( ) B 1, 0 , and C, where C is the point on the unit circle corresponding to an angle of ° 105 when it is drawn in standard position. Find the area of the triangle. State the answer in complete simplified form with a rationalized denominator. Challenge Problems Inscribed Circle For Problems 59–62, the lines that bisect each angle of a triangle meet in a single point O, and the perpendicular distance r from O to each side of the triangle is the same. The circle with center at O and radius r is called the inscribed circle of the triangle (see the figure). Q a O b c r r r R P C 2 C 2 B 2 A 2 A 2 B 2 59. Use the formula from Problem 57 with triangle OPQ to show that = r c A B C sin 2 sin 2 cos 2 51. Geometry Refer to the figure. If = OA 1, show that: (a) Area α α Δ = OAC 1 2 sin cos (b) Area β β Δ = OCB OB 1 2 sin cos 2 (c) Area α β ( ) Δ = + OAB OB 1 2 sin (d) α β = OB cos cos (e) α β α β α β ( ) + = + sin sin cos cos sin [Hint: Area Δ = Δ + Δ OAB OAC OCB Area Area .] 52. Geometry Refer to the figure, in which a unit circle is drawn. The line segment DB is tangent to the circle and θ is acute. (a) Express the area of ΔOBC in terms of θ sin and θ cos . (b) Express the area of ΔOBD in terms of θ sin and θ cos . (c) The area of the sector OBC of the circle is θ 1 2 , where θ is measured in radians. Use the results of parts (a) and (b) and the fact that Δ < < Δ OBC OBC OBD Area Area Area to show that θ θ θ < < 1 sin 1 cos y x u C O 1 1 21 21 B D 53. The Cow Problem* A cow is tethered to one corner of a square barn, 10 feet by 10 feet, with a rope 100 feet long. What is the maximum grazing area for the cow? See the figure. Barn Rope Ba A2 A3 A1 10 10 B C D A O 1 a b *Suggested by Professor Teddy Koukounas of Suffolk Community College, who learned of it from a farmer in Virginia.
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