1952 AMC 12 第 46 题

Question

一个新长方形的底边等于某个已知长方形的对角线与较长边之和，新长方形的高等于该对角线与较长边之差。新长方形的面积：

The base of a new rectangle equals the sum of the diagonal and the greater side of a given rectangle, while the altitude of the new rectangle equals the difference of the diagonal and the greater side of the given rectangle. The area of the new rectangle is:

大于已知长方形的面积

Greater than the area of the given rectangle

等于已知长方形的面积

Equal to the area of the given rectangle

等于以已知长方形较短边为边长的正方形面积

Equal to the area of a square with its side equal to the smaller side of the given rectangle

等于以已知长方形较长边为边长的正方形面积

Equal to the area of a square with its side equal to the greater side of the given rectangle

等于以已知长方形的对角线和较短边为两边的长方形面积

Equal to the area of a rectangle whose dimensions are the diagonal and shorter side of the given rectangle

答案：C

知识点：矩形勾股定理平方差

难度评级：1500

小提示：

设对角线长为

d

，较长边和较短边分别为

L,W

Let

d

be the diagonal and

L,W

the greater and smaller sides

大提示：

新长方形的面积为

(d+L)(d-L)=d^2-L^2

The new area is

(d+L)(d-L)=d^2-L^2

解答：

设较长边和较短边的长度分别为

L

和

W

，对角线长为

d

。新长方形的面积为

(d+L)(d-L)=d^2-L^2\text{。}

由勾股定理，

d^2=L^2+W^2

，所以新面积为

W^2

。这正是以原长方形较短边为边长的正方形面积。

因此，正确答案是 C。

Let the greater and smaller side lengths be

L

and

W,

and let the diagonal be

d.

The new rectangle’s area is

(d+L)(d-L)=d^2-L^2.

The Pythagorean theorem gives

d^2=L^2+W^2,

so the new area is

W^2.

This is the area of a square whose side equals the smaller side of the original rectangle.

Thus, the correct answer is C.

1952 AMC 12 第 46 题

答案：C

难度评级：1500

小提示：

大提示：

解答：

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