1959 AMC 12 第 31 题

Question

一个面积为

40

的正方形内接于半圆。若在同一半径的整圆中内接一个正方形，则其面积为：

A square, with an area of

40,

is inscribed in a semicircle. The area of a square that could be inscribed in the entire circle with the same radius is:

80

100

120

160

200

设第一个正方形边长为

s

，并将其下边放在半圆直径上

Let the first square have side

s

and place its lower side on the semicircle’s diameter

相对于圆心，一个上顶点的横坐标为

\frac{s}{2}

，纵坐标为

s

A top vertex has horizontal coordinate

\frac{s}{2}

and vertical coordinate

s

relative to the center

设半圆半径为

r

，正方形边长为

s

，其中

s^2=40

。正方形一个上顶点相对圆心的水平距离为

\frac{s}{2}

，竖直距离为

s

，所以

r^2=s^2+\left(\frac s2\right)^2=\frac54s^2=50\text{。}

内接于整圆的正方形对角线长为

2r

，故面积为

2r^2=100

。

因此，正确答案是 B。

Let the semicircle have radius

r

and the square have side

s,

where

s^2=40.

A top vertex of the square is

\frac{s}{2}

horizontally and

s

vertically from the center, so

r^2=s^2+\left(\frac s2\right)^2=\frac54s^2=50.

A square inscribed in the full circle has diagonal

2r,

hence area

2r^2=100.

Thus, the correct answer is B.

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