1959 AMC 12 第 15 题

Question

在一个直角三角形中，斜边的平方等于两条直角边乘积的两倍。该三角形的一个锐角为：

In a right triangle the square of the hypotenuse is equal to twice the product of the legs. One of the acute angles of the triangle is:

15^\circ

30^\circ

45^\circ

60^\circ

75^\circ

将已知关系与勾股定理结合

Combine the given relation with the Pythagorean theorem

比较

a^2+b^2

与

2ab

Compare

a^2+b^2

with

2ab

若两条直角边为

a,b

，斜边为

c

，则

a^2+b^2=c^2=2ab\text{。}

因此

(a-b)^2=0

，所以两条直角边相等。这个直角三角形是等腰三角形，其锐角均为

45^\circ

。

因此，正确答案是 C。

If the legs are

a,b

and the hypotenuse is

c,

then

a^2+b^2=c^2=2ab.

Hence

(a-b)^2=0,

so the legs are equal. The right triangle is isosceles and its acute angles are

45^\circ.

Thus, the correct answer is C.

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