AMC10 2006 B

AMC10 2006 B · Q8

AMC10 2006 B · Q8. It mainly tests Pythagorean theorem, Coordinate geometry.

A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle?

一个面积为40的正方形内接于一个半圆中，如图所示。半圆的面积是多少？

(A) $20\pi$ $20\pi$

(B) $25\pi$ $25\pi$

(D) $40\pi$ $40\pi$

(E) $50\pi$ $50\pi$

Answer

Correct choice: (B)

正确答案：（B）

Solution

The square has side length $\sqrt{40}$. Let $r$ be the radius of the semicircle. Then $$r^2 = (\sqrt{40})^2 + \left(\frac{\sqrt{40}}{2}\right)^2 = 40 + 10 = 50,$$ so the area of the semicircle is $\frac{1}{2}\pi r^2 = 25\pi$.

正方形边长为$\sqrt{40}$。设半圆半径为$r$。则 $$r^2 = (\sqrt{40})^2 + \left(\frac{\sqrt{40}}{2}\right)^2 = 40 + 10 = 50,$$ 因此半圆的面积为$\frac{1}{2}\pi r^2 = 25\pi$。

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