CHAPTER 7 Trigonometric Applications to Triangles Section 7.2: Area of a Triangle Finding Areas Finding Areas Formulas for the Area of a Triangle: 600 University of Houston Department of Mathematics
SECTION 7.2 Area of a Triangle MATH 1330 Precalculus 601
CHAPTER 7 Trigonometric Applications to Triangles Example: Solution: Area of a Segment of a Circle: 602 University of Houston Department of Mathematics
SECTION 7.2 Area of a Triangle MATH 1330 Precalculus 603
CHAPTER 7 Trigonometric Applications to Triangles Example: Solution: Additional Example 1: 604 University of Houston Department of Mathematics
SECTION 7.2 Area of a Triangle Solution: Additional Example 2: Solution: MATH 1330 Precalculus 605
CHAPTER 7 Trigonometric Applications to Triangles Additional Example 3: Solution: 606 University of Houston Department of Mathematics
SECTION 7.2 Area of a Triangle Additional Example 4: Solution: MATH 1330 Precalculus 607
CHAPTER 7 Trigonometric Applications to Triangles Additional Example 5: 608 University of Houston Department of Mathematics
SECTION 7.2 Area of a Triangle Solution: MATH 1330 Precalculus 609
CHAPTER 7 Trigonometric Applications to Triangles 610 University of Houston Department of Mathematics
Exercise Set 7.2: Area of a Triangle 1 Use the formula A bh to find the area of each of 2 the following triangles. (You may need to find the base and/or the height first, using trigonometric ratios or the Pythagorean Theorem.) Give exact values whenever possible. Otherwise, round answers to the nearest hundredth. Note: Figures may not be drawn to scale. 7. 9 in 57 o 1. 8 cm 8. 38 o 10 cm 7 m 2. 7 in In ABC below, AD BC. Use the diagram below to answer the following questions. A 3 in b h c 3. 3 ft 8 ft C D B a 4. 5. 6. 61 cm 6 cm 10 m 30 o 8 m 45 o 9. (a) Use ACD to write a trigonometric ratio that involves C, b, and h. (b) Using the equation from part (a), solve for h. (c) If a represents the base of ABC, and h represents the height, then the area K of 1 ABC is K ah. Substitute the equation 2 from part (b) into this equation to obtain a formula for the area of ABC that no longer contains h. 10. (a) Use ABD to write a trigonometric ratio that involves B, c, and h. (b) Using the equation from part (a), solve for h. (c) If a represents the base of ABC, and h represents the height, then the area K of 1 ABC is K ah. Substitute the equation 2 from part (b) into this equation to obtain a formula for the area of ABC that no longer contains h. MATH 1330 Precalculus 611
Exercise Set 7.2: Area of a Triangle Find the area of each of the following triangles. Give exact values whenever possible. Otherwise, round answers to the nearest hundredth. Note: Figures may not be drawn to scale. 16. 4 cm C 50 o 8 cm 11. B 6 in 120 o 10 in A T A C 17. J 12. 11 cm B 53 o 4.1 m A 30 o 10 cm C M 36 o 5.6 m A 13. L 18. 7.9 m 21 o S 9 m J 8 m 45 o K Q 3.5 m 125 o R 14. 15. D 7 ft 12 m 135 o E 5 ft G F Answer the following. Give exact values whenever possible. Otherwise, round answers to the nearest hundredth. 19. Find the area of an isosceles triangle with legs measuring 7 inches and base angles measuring 22.5. 20. Find the area of an isosceles triangle with legs measuring 10 cm and base angles measuring 75. D 28 o 8 m O 21. Find the area of GHJ, where G 120, h 8 cm, and j 15 cm. 22. Find the area of PQR, where R 47, p 7 in, and q 10 in. 612 University of Houston Department of Mathematics
Exercise Set 7.2: Area of a Triangle 23. Find the area of TUV, where T 28, U 81, t 12 m, and u 6.5 m. 24. Find the area of FUN, where F 92, N 28, f 9 cm, and n 8 cm. In the following questions, the radius of circle O is given, as well as the measure of central angle AOB. Find the area of the segment of circle O bounded by AB and AB. Give exact values whenever possible. Otherwise, round answers to the nearest hundredth. 25. (a) Find all solutions to sin x 0.5 for 0 x 180. (b) If the area of RST is 20 cm 2, r 16 cm and t 5 cm, find all possible measures for S. O A B 26. (a) Find all solutions to sin x 0.2 for 0 x 180. (b) If the area of PHL is 12 m 2, p 10 m and h 6m, find all possible measures for L. 27. If the area of TRG is 37 in 2, t 8 in and g 12.5 in, find all possible measures for R. 28. If the area of PSU is 20 2 cm 2, p 4 cm and s 5 cm, find all possible measures for U. 29. A regular octagon is inscribed in a circle of radius 12 in. Find the area of the octagon. 37. Radius: 8 cm Central Angle: 3 7 38. Radius: 7 in Central Angle: 39. Radius: 4 in Central Angle: 4 5 3 40. Radius: 6 cm Central Angle: 3 4 30. A regular hexagon is inscribed in a circle of radius 11 cm. Find the area of the hexagon. 31. A regular hexagon is circumscribed about a circle of radius 6 in. Find the area of the hexagon. 32. A regular hexagon is circumscribed about a circle of radius 10 cm. Find the area of the hexagon. 33. Find the area of an equilateral triangle with side length 4 ft. 34. Find the area of a regular hexagon with side length 10 cm. 35. Find the area of a regular decagon with side length 8 in. 36. Find the area of a regular octagon with side length 12 m. MATH 1330 Precalculus 613