1998 AMC 12 第 8 题

Question

一个边长为

1

的正方形被分成两个全等梯形和一个五边形，三个图形面积相等；分割线由正方形中心连接三条边上的点而成，如图所示。求每个梯形较长平行边的长度

x

。

A square with sides of length

1

is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find

x,

the length of the longer parallel side of each trapezoid.

\frac35

\frac23

\frac34

\frac56

\frac78

答案：D

知识点：trapezoid area equal areas

难度评级：1540

小提示：

三个区域的面积均为

\frac{1}{3}

Each of the three regions has area

\frac{1}{3}

大提示：

每个梯形的高为

\frac{1}{2}

，两条平行边长分别为

x

和

\frac{1}{2}

A trapezoid has height

\frac{1}{2}

and parallel sides

x

and

\frac{1}{2}

解答：

每个梯形的面积为

\frac{1}{3}

。它的高为

\frac{1}{2}

，两条平行边长为

x

和

\frac{1}{2}

。因此

\frac12\cdot\frac12\left(x+\frac12\right)=\frac13\text{，}

解得

x=\frac{5}{6}

。所以正确答案是 D。

Each trapezoid has area

\frac{1}{3}.

Its height is

\frac{1}{2},

and its parallel sides have lengths

x

and

\frac{1}{2}.

Therefore

\frac12\cdot\frac12\left(x+\frac12\right)=\frac13,

which gives

x=\frac{5}{6}.

Thus D is correct.

1998 AMC 12 第 8 题

答案：D

难度评级：1540

小提示：

大提示：

解答：

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