AMC 8 2020 Problem 18
AMC8 2020 年第 18 题
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Rectangle ABCD is inscribed in a semicircle with diameter , as shown in the figure. Let DA=16, and let FD=AE=9. What is the area of ABCD?
如图所示，矩形 内接于以 为直径的半圆中。已知 DA=16，FD=AE=9。那么 的面积是多少？
- A.240
- B.248
- C.256
- D.264
- E.272
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
Let 𝑂 O be the center of the circle. Then 𝐷 𝑂 = 𝑂 𝐴 = 8 DO=OA=8 and radius 𝑂 𝐸 = 𝑂 𝐴 + 𝐴 𝐸 = 8 + 9 = 17 OE=OA+AE=8+9=17. Draw radius 𝑂 𝐵 ‾ OB which also has length 17 17. Then △ 𝑂 𝐴 𝐵 △OAB is a right triangle with hypotenuse 17 17 and base leg 8 8. By the Pythagorean Theorem 𝐴 𝐵 = 1 7 2 − 8 2 = 289 − 64 = 225 = 15 AB= 17 2 −8 2 = 289−64 = 225 =15. The area of rectangle 𝐴 𝐵 𝐶 𝐷 ABCD is 𝐷 𝐴 ⋅ 𝐴 𝐵 DA⋅AB = 16 ⋅ 15 16⋅15 = 240 240. Answer: 𝐴 A .
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