AMC8 2015

AMC8 2015 · Q2

AMC8 2015 · Q2. It mainly tests Area & perimeter, Polygons.

Point O is the center of the regular octagon ABCDEFGH, and X is the midpoint of side AB. What fraction of the area of the octagon is shaded?

点O是正八边形ABCDEFGH的中心，X是边AB的中点。阴影部分占八边形面积的几分之几？

(A) $\frac{11}{32}$ $\frac{11}{32}$

(B) $\frac{3}{8}$ $\frac{3}{8}$

(D) $\frac{7}{16}$ $\frac{7}{16}$

(E) $\frac{15}{32}$ $\frac{15}{32}$

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): The octagon can be divided into 8 congruent triangles, 3 of which are BOC, ACOD, and ADOE. The area of triangle XOB is half the area of triangle BOC, so the fraction of the area of the octagon that is shaded is $3\cdot \frac{1}{8} + \frac{1}{2}\cdot \frac{1}{8} = \frac{7}{16}$.

解答（D）：这个八边形可以被分成 8 个全等的三角形，其中 3 个分别是 BOC、ACOD 和 ADOE。三角形 XOB 的面积是三角形 BOC 面积的一半，因此八边形被阴影部分所占的面积比例为 $3\cdot \frac{1}{8} + \frac{1}{2}\cdot \frac{1}{8} = \frac{7}{16}$。

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