SECTION 8.4 Area of a Triangle 581 In Problems 31–36, use the results of Problem 29 or 30 to find the area of each triangle. Round answers to two decimal places. 31. = ° = ° = A B a 40 , 20 , 2 32. = ° = ° = A C a 50 , 20 , 3 33. = ° = ° = B C b 70 , 10 , 5 34. = ° = ° = A B c 70 , 60 , 4 35. = ° = ° = A C c 110 , 30 , 3 36. = ° = ° = B C b 10 , 100 , 2 37. Area of a Segment Find the area of the segment (shaded in blue in the figure) of a circle whose radius is 8 feet, formed by a central angle of ° 70 . [Hint: Subtract the area of the triangle from the area of the sector to obtain the area of the segment.] 38. Area of a Segment Find the area of the segment of a circle whose radius is 5 inches, formed by a central angle of ° 40 . 39. Cost of a Triangular Lot The dimensions of a triangular lot are 100 feet by 50 feet by 75 feet. If the price of the land is $300 per square foot, how much does the lot cost? 40. Amount of Material to Make a Tent A cone-shaped tent is made from a circular piece of canvas 24 feet in diameter by removing a sector with central angle ° 100 and connecting the ends. What is the surface area of the tent? 41. Fighter Jet Design The Eurofighter Typhoon has a canarddelta wing design that contains a large triangular main wing. Use the dimensions shown to approximate the area of one of the main wings. 8.75 m 8.05 m 4.55 m Credit: sam-whitfield1/Shutterstock 42. Property Area A lot for sale in a subdivision has the shape of the quadrilateral shown in the figure. Find the area of the lot to the nearest square foot. 43. Dimensions of Home Plate The dimensions of home plate at any major league baseball stadium are shown. Find the area of home plate. 12 in. 12 in. 17 in. 8.5 in. 8.5 in. 44. Computing Areas See the figure. Find the area of the shaded region enclosed in a semicircle of diameter 10 inches. The length of the chord PQ is 8 inches. [Hint: Triangle PQR is a right triangle.] 45. Geometry See the figure, which shows a circle of radius r with center at O. Find the area K of the shaded region as a function of the central angle θ. 46. Approximating the Area of a Lake To approximate the area of a lake, a surveyor walks around the perimeter of the lake, taking the measurements shown in the figure. Using this technique, what is the approximate area of the lake? [Hint: Use the Law of Cosines on the three triangles shown, and then find the sum of their areas.] 80 ft 20 ft 35 ft 40 ft 45 ft 1008 158 47. The Flatiron Building Completed in 1902 in New York City, the Flatiron Building is triangular shaped and bounded by 22nd Street, Broadway, and 5th Avenue. The building measures approximately 87 feet on the 22nd Street side,190 feet on the Broadway side,and173 feet on the 5thAvenue side.Approximate the ground area covered by the building. Source: Sarah Bradford Landau and Carl W. Condit, Rise of the New York Skyscraper: 1865–1913. New Haven, CT: Yale University Press, 1996. 48. Bermuda Triangle The Bermuda Triangle is roughly defined by Hamilton, Bermuda; San Juan, Puerto Rico; and Fort Lauderdale, Florida. The distances from Hamilton to Fort Lauderdale, Fort Lauderdale to San Juan, and San Juan to Hamilton are approximately 1028, 1046, and 965 miles, respectively. Ignoring the curvature of Earth, approximate the area of the Bermuda Triangle. Source: www.worldatlas.com 49. Bretschneider’s Formula There is a Heron-type formula that can be used to find the area of a general quadrilateral. θ ( )( )( )( ) = − − − − − K sasbscsd abcd cos2 where a, b, c, and d are the side lengths, θ is half the sum of two opposite angles, and s is half the perimeter. Show that if a triangle is considered a quadrilateral with one side equal to 0, Bretschneider’s Formula reduces to Heron’s Formula. 50. (a) Show that the area of a regular dodecagon (12-sided polygon)isgivenby π π = = K a K r 3 cot 12 or 12 tan 12 , 2 2 where a is the length of one of the sides and r is the radius of the inscribed circle. (b) Given that each interior angle of a regular n-sided polygon ( ) ≥ n 3 measures π ( ) − ⋅ n n 2 , generalize these formulas for any regular polygon. 138 ft 38 ft 86 ft 140 ft 858 8 Q R P 10 r O u 70° 8 ft
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