11-2 MAIN IDEA Find the areas of triangles and trapezoids. Area of Triangles and Trapezoids 1. 24 sq units 2. They are congruent. • Draw a parallelogram with a base of 6 units and a height of 4 units. • Draw a diagonal as shown. Math Online • Cut out the parallelogram. glencoe.com 1. What is the area of the parallelogram? • Concepts In Motion • Extra Examples • Personal Tutor • Self-Check Quiz 2. Cut along the diagonal. What is true about the triangles formed? 3. What is the area of each triangle? 12 sq units 4. If the area of a parallelogram is bh, then write an expression for the area A of each of the two congruent triangles that form the parallelogram. A = 1 bh _ 2 You can find the area of a triangle by using the base and height. The base of a triangle can be any of its sides. h The height is the perpendicular distance from the vertex opposite the base to the line containing the base. b Area of a Triangle Words Symbols The area A of a triangle equals half the product of its base b and height h. Key Concept Model _1 A = bh h b 2 Find the Area of a Triangle 1 Find the area of the triangle shown. _ 1 Estimate A = (10)(7) or 35 2 1 A=_ bh 2 1 ( )( ) A=_ 10 6.5 2 A = 32.5 Area of a triangle Replace b with 10 and h with 6.5. Multiply. 6.5 m 10 m The area of the triangle is 32.5 square meters. Check for Reasonableness 32.5 ≈ 35 ✔ 578 Chapter 11 Measurement: Two- and Three-Dimensional Figures 578_0578_0582_CH11_L2_874046 578 9/26/07 7:32:52 PM Find the area of each triangle. Round to the nearest tenth if necessary. a. b. 77 ft 2 30 in 2 8 in. 11 ft 1 7_ in. 14 ft 2 A trapezoid has two bases, b 1 and b 2. The bases are always the two sides that are parallel. The height of a trapezoid is the perpendicular distance between the bases. Area of a Trapezoid Reading Math Subscripts Read b 1 as b sub 1. Read b 2 as b sub 2. The subscripts mean that b 1 and b 2 represent different variables. Words Key Concept The area A of a trapezoid equals half the product of the height h and the sum of the bases b 1 + b 2. b1 Model h _1 b2 A = h(b 1 + b 2) Symbols 2 Find the Area of a Trapezoid 2 Find the area of the trapezoid. 5 in. The bases are 5 inches and 12 inches. The height is 7 inches. 1 A=_ h(b1 + b2 ) 2 _ A = 1 (7 ) (5 + 12 ) 2 1 ( )( ) A=_ 7 17 2 A = 59.5 7 in. Area of a trapezoid 12 in. Replace h with 7, b 1 with 5, and b 2 with 12. Add 5 and 12. Multiply. The area of the trapezoid is 59.5 square inches. Find the area of each trapezoid. Round to the nearest tenth if necessary. c. 2.5 m 14.6 m 2 d. 1 ft 0.2 ft 2 0.3 ft 4m 4.8 m 0.5 ft Lesson 11-2 Area of Triangles and Trapezoids 579_0578_0582_CH11_L2_874046 579 579 9/26/07 7:32:58 PM 3 GEOGRAPHY The shape of Iron County, Michigan, resembles a trapezoid. Find the approximate area of this county. Keweenaw Houghton Ontonagon Gogebic 22 mi Baraga 42 mi Iron Marquette Alger 34 mi Dickinson Luce Schoolcraft Chippewa Mackinac Delta Menominee 1 A=_ h(b 1 + b 2) 2 _ A = 1 (42)(22 + 34) 2 _ A = 1 (42)(56) 2 A = 1,176 Area of a trapezoid Replace h with 42, b 1 with 22, and b 2 with 34. Add 22 and 34. Multiply. The area of Iron County is approximately 1,176 square miles. 280 mi e. GEOGRAPHY The shape of the state ARKANSAS of Arkansas resembles a trapezoid. Find the approximate area of Arkansas. 235 mi Little Rock 210 mi Examples 1, 2 (pp. 578–579) Find the area of each figure. Round to the nearest tenth if necessary. 1. 2. 3. 16.5 m 3 in. 4 in. Example 3 (p. 580) 580 8 ft 15.6 ft 12.8 m 4. HOCKEY In the National Hockey League, goaltenders can play the puck behind the goal line only in a trapezoid-shaped area, as shown at the right. Find the area of the trapezoid. 7 ft 18 ft 11 ft 28 ft Chapter 11 Measurement: Two- and Three-Dimensional Figures 578_582_C11_L2_892329.indd 580 4/8/10 2:01:34 PM HOMEWORK For Exercises 5, 6, 9 7, 8, 10 11, 12 HELP See Examples 1 2 3 Find the area of each figure. Round to the nearest tenth if necessary. 5. 147 in 2 6. 38.4 mm 2 8 mm 14 in. 21 in. 7. 1.1 cm 2 cm 9.6 mm 3.4 cm 4.5 cm 2 Exercise Levels A: 5–12 B: 13–20 C: 21–23 17.75 m 8. 9. 183.7 in 2 112 m 2 10 ft 1 22 in. 8m 15 ft 10. 8 2 ft 10.25 m 16.7 in. 23 ft 161.5 ft 2 11. GEOGRAPHY The shape of Idaho is roughly triangular with a base of 380 miles and a height of 500 miles. Find the approximate area of Idaho. about 95,000 mi 2 12. ALGEBRA Find the area of a trapezoid with bases 13 inches and 15 inches and a height of 7 inches. 98 in 2 500 mi IDAHO 380 mi ALGEBRA Find the height of each figure. 13. 174 ft x ft 264 yd 14. 125 ft x yd 2 A = 11,500 ft 184 ft 130 yd 2 A = 29,185 yd 145 yd 185 yd Draw and label each figure. Then find the area. 15–17. See margin. 15. a triangle with no right angles and an area less than 12 square centimeters 16. a trapezoid with a right angle and an area greater than 40 square meters 17. a trapezoid with no right angles and an area less than 25 square feet 23 in. ★ 18. TENTS A play tent is shown at the right. How much fabric was used to make the front and back of the play tent? 1,904 in 2 32 in. 36.5 in. LANDSCAPING For Exercises 19 and 20, use the diagram that shows the lawn that surrounds an office building. ★ 19. What is the area of the lawn? 6,500 ft 2 EXTRA PRACTICE See pages 697, 714. 20. If one bag of grass seed covers 2,000 square feet, how many bags are needed to seed the lawn? 4 bags 100 ft 50 ft 80 ft 62 ft 140 ft Lesson 11-2 Area of Triangles and Trapezoids 581_0578_0582_CH11_L2_874046 581 581 9/26/07 7:33:07 PM H.O.T. Problems 21. Sample answer: The ratio of the areas is the square of the ratio of the bases. 21. CHALLENGE Triangle ABC has a base of 4 units and a height of 8 units. Triangle DEF has twice the base and height. Describe how the ratio of the bases is related to the ratio of the areas. h to explain how to estimate the height h of the trapezoid shown if the area is 235.5 in 2. See margin. 23. 25. SHORT RESPONSE Randy was hired to J F 12 ft x ft 18 ft G 26.75 in. WR ITING IN MATH Describe the relationship between the area of a parallelogram and the area of a triangle with the same height and base. The area of a triangle is half the area of a parallelogram with the same base and height, because two of these triangles make up the parallelogram. 24. FGH and JKM are similar. H 19.95 in. 22. REASONING Apply what you know about rounding M 24 ft put in sod on a piece of land shaped like a trapezoid as shown. How many square feet of sod are needed? 11,000 ft2 K Which choice shows the equations that can be used to find the area of FGH? C 18 12 1( A _ =_ and then _ 18x) 50 ft 120 ft 80 ft 2 24 x 18 x B _ =_ and then 18x 24 12 18 x 1( C _ =_ and then _ 18x) 2 24 12 18 12 D _ =_ and then 18x 24 x 140 ft 225 ft 26. MEASUREMENT Find the area of a parallelogram having a base of 2.3 inches and a height of 1.6 inches. Round to the nearest tenth. (Lesson 11-1) 3.7 in 2 ★27. GEOMETRY Graph JLK with vertices J(-1, -4), L(3, -2) and K(1, 1), and its reflection over the x-axis. Write the ordered pairs for the vertices of the new figure. (Lesson 10-10) See margin. Find each number. Round to the nearest tenth if necessary. 28. What number is 56% of 600? 336 30. 72 is 45% of what number? 160 (Lesson 7-2) 29. 24.5 is what percent of 98? 25% 31. 62.5% of 250 is what number? 156.3 PREREQUISITE SKILL Use a calculator to find each product to the nearest tenth. (Lesson 1-4) 32. π · 13 582 40.8 33. π · 29 91.1 34. π · 16 2 804.2 35. π · 4.8 2 72.4 Chapter 11 Measurement: Two- and Three-Dimensional Figures 582_0578_0582_CH11_L2_874046 582 9/26/07 7:33:13 PM

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