1995 AMC 12 第 22 题

Question

从一张长方形纸上剪去一个三角形角，形成一个五边形。五边形的五条边长为

13

、

19

、

20

、

25

、

31

，但这不一定是它们沿五边形周界的顺序。五边形的面积是

A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths

13,

19,

20,

25

and

31,

although this is not necessarily their order around the pentagon. The area of the pentagon is

459

600

680

720

745

答案：E

知识点：rectangles Pythagorean theorem area subtraction

难度评级：1910

小提示：

五边形的两条边就是原长方形的两条完整边长

Two pentagon sides are the original rectangle’s full side lengths

大提示：

在这些长度中寻找两个差，使它们与另一个给定长度构成直角三角形

Look for two differences among the lengths that form the legs of a right triangle with a third listed length

解答：

长方形的完整边长必须大于同一边上剪后剩余部分的长度。唯一能使两个差与剩下的切边构成直角三角形的安排，是边长为

25

和

31

的长方形，对应剩余长度为

20

和

19

。被剪去的角所成直角三角形的两条直角边为

25-20=5

和

31-19=12

，斜边正好是给定边长

13

。因此五边形面积为

\begin{aligned} 25\cdot31-\frac12(5)(12) &=775-30\\ &=745 \end{aligned}\text{。}

所以正确答案是 E。

The full rectangle sides must be longer than the two remnants on those same sides. The only assignment whose two differences and remaining cut side form a right triangle is the rectangle

25

by

31,

with remnants

20

and

19.

The removed corner then has legs

25-20=5

and

31-19=12,

whose hypotenuse is the listed side

13.

Thus the pentagon’s area is

\begin{aligned} 25\cdot31-\frac12(5)(12) &=775-30\\ &=745. \end{aligned}

Hence the correct answer is E.

1995 AMC 12 第 22 题

答案：E

难度评级：1910

小提示：

大提示：

解答：

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