10-2 Areas of Trapezoids, Rombuses, and Kites Common Core State Standards G-MG.A.1 Use geometric sapes, teir measures, and teir properties to describe objects. MP 1, MP 3, MP 4, MP 6 Objective To find te area of a trapezoid, rombus, or kite Rearranging figures into familiar sapes is an example of te Solve a Simpler Problem strategy. Draw a trapezoid on a seet of grap paper. Label te bases b 1 and b 2. Draw its midsegment. Cut out te trapezoid, and ten cut it along te midsegment. Rotate te top part of te trapezoid 180 so tat b 1 and b 2 now form one long base. How can you use tis new figure to find te area of te trapezoid? Explain your reasoning. b 1 b 2 Lesson L VocabularyV eigt of a trapezoid MATHEMATICAL PRACTICES Essential Understanding You can find te area of a trapezoid wen you know its eigt and te lengts of its bases. Te eigt of a trapezoid is te perpendicular distance between te bases. Teorem 10-4 Area of a Trapezoid Te area of a trapezoid is alf te product of te eigt and te sum of te bases. A = 1 2 (b 1 + b 2 ) b 1 b 2 Problem 1 Area of a Trapezoid Wic borders of Nevada can you use as te bases of a trapezoid? Te two parallel sides of Nevada form te bases of a trapezoid. Geograpy Wat is te approximate area of Nevada? A = 1 2 (b 1 + b 2 ) Use te formula for area of a 205 mi trapezoid. = 1 2 (309)(205 + 511) Substitute 309 for, 205 for b 1, and 511 for b 2. = 110,622 Simplify. 309 mi Reno Carson City 511 mi Te area of Nevada is about 110,600 mi 2. Las Vegas 1. Wat is te area of a trapezoid wit eigt 7 cm and bases 12 cm and 15 cm? Lesson 10-2 Areas of Trapezoids, Rombuses, and Kites 623
Problem 2 Finding Area Using a Rigt Triangle Wat is te area of trapezoid PRS? S 5 m R How are te sides related in a 30-60 -90 triangle? Te lengt of te ypotenuse is 2 times te lengt of te sorter leg, and te longer leg is 13 times te lengt of te sorter leg. You can draw an altitude tat divides te trapezoid into a rectangle and a 30-60 -90 triangle. Since te opposite sides of a rectangle are congruent, te longer base of te trapezoid is divided into segments of lengts 2 m and 5 m. = 2 13 longer leg = sorter leg # 13 A = 1 2 (b 1 + b 2 ) Use te trapezoid area formula. = 1 2 (213 )(7 + 5) Substitute 213 for, 7 for b 1, and 5 for b 2. = 1213 Simplify. P P 60 7 m S 5 m 60 2 m 5 m R Te area of trapezoid PRS is 121 2. 2. Reasoning In Problem 2, suppose decreases so tat m P = 45 wile angles R and and te bases stay te same. Wat is te area of trapezoid PRS? Essential Understanding You can find te area of a rombus or a kite wen you know te lengts of its diagonals. Teorem 10-5 Area of a Rombus or a Kite Te area of a rombus or a kite is alf te product of te lengts of its diagonals. d 1 d 2 d 1 A = 1 2 d 1 d 2 d 2 Rombus Rombus Kite Kite Problem 3 Finding te Area of a Kite Do you need to know te side lengts of te kite to find its area? No. You only need te lengts of te diagonals. Wat is te area of kite KLMN? Find te lengts of te two diagonals: KM = 2 + 5 = 7 m and LN = 3 + 3 =. Use te formula for area of a kite. A = 1 2 d 1 d 2 = 1 2 (7)(6) Substitute 7 for d 1 and 6 for d 2. = 21 Simplify. K 2 m N L 5 m M Te area of kite KLMN is 21 m 2. 3. Wat is te area of a kite wit diagonals tat are 12 in. and 9 in. long? 624 Capter 10 Area
Problem 4 Finding te Area of a Rombus How can you find te lengt of AB? AB is a leg of rigt ABC. You can use te Pytagorean Teorem, a 2 + b 2 = c 2, to find its lengt. Car Pooling Te Hig Occupancy Veicle (HOV) lane is marked by a series of diamonds, or rombuses painted on te pavement. Wat is te area of te HOV lane diamond sown at te rigt? ABC is a rigt triangle. Using te Pytagorean Teorem, AB = 26.5 2-2.5 2 = 6. Since te diagonals of a rombus bisect eac oter, te diagonals of te HOV lane diamond are 5 ft and 12 ft. A = 1 2 d 1 d 2 Use te formula for area of a rombus. = 1 2 (5)(12) Substitute 5 for d 1 and 12 for d 2. = 30 Simplify. Te area of te HOV lane diamond is 30 ft 2. A 6.5 ft C B 2.5 ft 4. A rombus as sides 10 cm long. If te longer diagonal is 16 cm, wat is te area of te rombus? Lesson Ceck Do you know HOW? Find te area of eac figure. 1. 4 m 2. 3. 3 ft 4. 5 ft 5. 6. 20 m 15 in. 18 in. 27 in. 12 in. 12 in. 2 cm 2 cm 1 cm Do you UNDERSTAND? 7. Vocabulary Can a trapezoid and a parallelogram wit te same base and eigt ave te same area? Explain. b 8. Reasoning Do you need to know all te side lengts to find te area of a trapezoid? 9. Reasoning Can you find te area of a rombus if you only know te lengts of its sides? Explain. 10. Reasoning Do you need to know te lengts of te sides to find te area of a kite? Explain. b MATHEMATICAL PRACTICES Lesson 10-2 Areas of Trapezoids, Rombuses, and Kites 625
Practice and Problem-Solving Exercises MATHEMATICAL PRACTICES A Practice Find te area of eac trapezoid See Problem 1. 11. 21 in. 12. 24. 13. 16 in. 8.5 cm 9 ft 18 ft 6 ft 9.7 cm 38 in. 14. Find te area of a trapezoid wit bases 12 cm and 18 cm and eigt 10 cm. 15. Find te area of a trapezoid wit bases 2 ft and 3 ft and eigt 1 3 ft. 16. Geograpy Te border of Tennessee resembles a trapezoid wit bases 340 mi and 440 mi and eigt 1i. Estimate te area of Tennessee by finding te area of te trapezoid. Find te area of eac trapezoid. If your answer is not an integer, leave it in simplest radical form. See Problem 2. 17. 5 ft 18. 19. 3 ft 8 ft 6 ft 60 15 ft 8 m Find te area of eac kite. 20. 2 in. 21. 22. 2 m 8 in. 8 in. 4 m 8 in. 6 ft 4 ft 4 ft See Problem 3. Find te area of eac rombus. 23. 24. 10 in. 25. 8 in. 30 ft 20 ft See Problem 4. 5 m B Apply 26. Tink About a Plan A trapezoid as two rigt angles, 12-m and 18-m bases, and an 8-m eigt. Sketc te trapezoid and find its perimeter and area. Are te rigt angles consecutive or opposite angles? How does knowing te eigt elp you find te perimeter? 626 Capter 10 Area
27. Metallurgy Te end of a gold bar as te sape of a trapezoid wit te measurements sown. Find te area of te end. 28. Open-Ended Draw a kite. Measure te lengts of its diagonals. Find its area. Find te area of eac trapezoid to te nearest tent. 29. 30. 8 ft 31. 4 cm 1 cm 30 9 ft 6.9 cm 4.4 cm 9.2 cm 1.7 m 45 2.1 m 0.9 m Coordinate Geometry Find te area of quadrilateral RST. 32. y R 33. y R 34. 2 4 S 2 2 4 x 2 T 2 S 4 x T y S R 2 2 O 2 T 2 x 35. Wat is te area of te kite at te rigt? 90 m 2 135 m 2 108 m 2 21 2 36. a. Coordinate Geometry Grap te lines x = 0, x = 6, y = 0, and y = x + 4. b. Wat type of quadrilateral do te lines form? c. Find te area of te quadrilateral. Find te area of eac rombus. Leave your answer in simplest radical form. 9V2 m 45 37. 38. 39. 45 60 8 in. 30 4 m 40. Visualization Te kite as diagonals d 1 and d 2 congruent to te sides of te rectangle. Explain wy te area of te kite is 1 2 d 1 d 2. 41. Draw a trapezoid. Label its bases b 1 and b 2 and its eigt. Ten draw a diagonal of te trapezoid. a. Write equations for te area of eac of te two triangles formed. b. Writing Explain ow you can justify te trapezoid area formula using te areas of te two triangles. d 1 d 2 Lesson 10-2 Areas of Trapezoids, Rombuses, and Kites 627
C Callenge 42. Algebra One base of a trapezoid is twice te oter. Te eigt is te average of te two bases. Te area is 324 cm 2. Find te eigt and te bases. (Hint: Let te smaller base be x.) y 43. Sports Ty wants to paint one side of 1 te skateboarding ramp e built. Te y 0.25x ramp is 4 m wide. Its surface is modeled by te equation y = 0.25x 2. Use te trapezoids and triangles O 2 1 sown to estimate te area to be 1 2 painted. 44. In trapezoid ABCD at te rigt, AB } DC. Find te area of ABCD. A 15 in. B x 20 in. 135 D 30 C Standardized Test Prep SAT/ACT Sort Response 45. Te area of a kite is 120 cm 2. Te lengt of one diagonal is 20 cm. Wat is te lengt of te oter diagonal? 12 cm 20 cm 24 cm 48 cm 46. ABC XYZ. AB = 6, BC = 3, and CA = 7. Wic of te following are NOT possible dimensions of XYZ? XY = 3, YZ = 1.5, ZX = 3.5 XY = 10, YZ = 7, ZX = 11 XY = 9, YZ = 4.5, ZX = 10.5 XY = 18, YZ = 9, ZX = 21 47. Draw an angle. Construct a congruent angle and its bisector. Mixed Review 48. Find te area of a rigt isosceles triangle tat as one leg of lengt 12 cm. 49. A rigt isosceles triangle as area 112.5 ft 2. Find te lengt of eac leg. 50. Find te measure of an interior angle of a regular nonagon. See Lesson 10-1. See Lesson 6-1. Get Ready! To prepare for Lesson 10-3, do Exercises 51 53. Find te area of eac regular polygon. Leave radicals in simplest form. 51. 52. 53. 10 cm 10 ft See Lesson 8-2. 628 Capter 10 Area