AMC 8 2011 Problem 20
AMC8 2011 年第 20 题
四边形与多边形★★★☆☆
Quadrilateral ABCD is a trapezoid, AD = 15 , AB = 50 , BC = 20 , and the altitude is 12 . What is the area of the trapezoid?
四边形 ABCD 是个梯形，其中 AD＝15，AB＝50，BC＝20，且高为 12.那么这个梯形的面积是多少?
- A.600
- B.650
- C.700
- D.750
- E.800
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Let 𝐸 E and 𝐹 F be the feet of the perpendiculars from 𝐴 A and 𝐵 B to 𝐷 𝐶 ‾ DC. In right △ 𝐴 𝐸 𝐷 △AED, 𝐷 𝐸 2 = 1 5 2 − 1 2 2 = 225 − 144 = 81 DE 2 =15 2 −12 2 =225−144=81, so 𝐷 𝐸 = 9 DE=9. In right △ 𝐵 𝐹 𝐶, 𝐹 𝐶 2 = 2 0 2 − 1 2 2 = 400 − 144 = 256 △BFC,FC 2 =20 2 −12 2 =400−144=256, so 𝐹 𝐶 = 16 FC=16. Right △ 𝐴 𝐸 𝐷 △AED has area 1 2 ⋅ 9 ⋅ 12 = 54 2 1 ⋅9⋅12=54, right △ 𝐵 𝐹 𝐶 △BFC has area 1 2 ⋅ 16 ⋅ 12 = 96 2 1 ⋅16⋅12=96, and rectangle 𝐴 𝐵 𝐹 𝐸 ABFE has area 50 ⋅ 12 = 600 50⋅12=600. The trapezoid 𝐴 𝐵 𝐶 𝐷 ABCD has area 54 + 96 + 600 = 750 54+96+600=750. OR OR Begin as in the first solution and note that 𝐷 𝐶 = 𝐷 𝐸 + 𝐸 𝐹 + 𝐹 𝐶 = 9 + 50 + 16 = DC=DE+EF+FC=9+50+16= 75. Then the area of trapezoid is 1 2 ( 𝐴 𝐵 + 𝐷 𝐶 ) ⋅ 𝐴 𝐸 = 1 2 ( 50 + 75 ) ⋅ 12 = 125 ⋅ 6 = 750 2 1 (AB+DC)⋅AE= 2 1 (50+75)⋅12=125⋅6=750. Answer: 𝐷 D .
继续练习