AMC8 2005

AMC8 2005 · Q23

AMC8 2005 · Q23. It mainly tests Triangles (properties), Circle theorems.

Isosceles right triangle ABC encloses a semicircle of area $2\pi$. The circle has its center O on hypotenuse AB and is tangent to sides AC and BC. What is the area of triangle ABC?

等腰直角三角形ABC内有一个面积为$2\pi$的半圆。圆心O在斜边AB上，且该圆与边AC和BC相切。三角形ABC的面积是多少？

(A) 6 6

(B) 8 8

(D) 10 10

(E) $4\pi$ 4π

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) Reflect the triangle and the semicircle across the hypotenuse $AB$ to obtain a circle inscribed in a square. The circle has area $4\pi$. The radius of a circle with area $4\pi$ is $2$. The side length of the square is $4$ and the area of the square is $16$. So the area of the triangle is $8$.

（B）将三角形和半圆关于斜边 $AB$ 反射，可得到一个内接于正方形的圆。该圆的面积为 $4\pi$。面积为 $4\pi$ 的圆的半径是 $2$。正方形的边长为 $4$，面积为 $16$。因此三角形的面积为 $8$。

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