1961 AMC 12 第 14 题

Question

一个菱形的一条对角线长度是另一条的两倍。用

K

表示该菱形的边长，其中

K

是以平方英寸为单位的菱形面积。

A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus in terms of

K,

where

K

is the area of the rhombus in square inches.

\sqrt K

\dfrac12\sqrt{2K}

\dfrac13\sqrt{3K}

\dfrac14\sqrt{4K}

以上都不正确

none of these are correct

设两条对角线为

d

和

2d

，再使用菱形面积公式

Let the diagonals be

d

and

2d

, then use the rhombus area formula

两条半对角线构成直角三角形的两条直角边，其斜边就是菱形的一条边

Half-diagonals form the legs of a right triangle whose hypotenuse is a side

设两条对角线为

d

和

2d

。则

K=\frac{d(2d)}{2}=d^2

。一条边的长度为

\begin{aligned} \sqrt{\left(\frac d2\right)^2+d^2} &=\frac{\sqrt5}{2}d\\ &=\frac12\sqrt{5K} \end{aligned}\text{。}

前四个选项中没有这个结果。

所以，正确答案是 E。

Let the diagonals be

d

and

2d.

Then

K=\frac{d(2d)}{2}=d^2.

A side has length

\begin{aligned} \sqrt{\left(\frac d2\right)^2+d^2} &=\frac{\sqrt5}{2}d\\ &=\frac12\sqrt{5K}. \end{aligned}

This is not among the first four choices.

Thus, the correct answer is E.

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