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：

其他年份的第 46 题