1950 AMC 12 第 9 题

Question

半径为

r

的半圆内所能内接的最大三角形的面积为：

The area of the largest triangle that can be inscribed in a semicircle whose radius is

r

is:

r^2

r^3

2r^2

2r^3

\dfrac12r^2

最长的底边是半圆的直径

The longest possible base is the diameter of the semicircle

以直径为底边时，最大高等于半径

With the diameter as base, the greatest possible altitude is the radius

半圆内三角形的底边至多为直径

2r

，高至多为

r

。因此，其面积至多为

\frac12(2r)(r)=r^2\text{。}

取直径为底边、半圆的最高点为第三个顶点时，可以达到这个上界。

因此，正确答案是 A。

A triangle in the semicircle has base at most the diameter

2r

and altitude at most

r.

Therefore its area is at most

\frac12(2r)(r)=r^2.

This bound is attained by using the diameter as the base and the topmost point of the semicircle as the third vertex.

Thus, the correct answer is A.

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