1957 AMC 12 第 49 题

Question

一个梯形的两条平行边长为

3

与

9

，两条非平行边长为

4

与

6

。一条平行于底边的直线将梯形分成两个周长相等的梯形。每条非平行边被分割的两段之比为：

The parallel sides of a trapezoid are

3

and

9.

The non-parallel sides are

4

and

6.

A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is:

4:3

3:2

4:1

3:1

6:1

答案：C

知识点：梯形比与比例周长

难度评级：1830

小提示：

平行于底边的线段按相同比例分割两条腰

A segment parallel to the bases divides both legs in the same fraction

大提示：

设两条腰的上段分别为

4t

与

6t

；令两个周长相等时，公共分割线段会相消

Let the upper leg segments be

4t

and

6t

; the common dividing segment cancels when the two perimeters are equated

解答：

设长度为

4

与

6

的两条腰的上段分别为

4t

与

6t

。分割线段在两个周长中各出现一次，可以相消。因此，由周长相等可得

\begin{aligned} 3+4t+6t &=9+4(1-t)\\ &\quad+6(1-t) \end{aligned}\text{。}

所以

20t=16

，从而

t=\frac{4}{5}

。每条腰被分割的两段之比为

t:(1-t)=\frac45:\frac15=4:1\text{。}

因此，正确答案是 C。

Let the upper pieces of the legs of lengths

4

and

6

be

4t

and

6t,

respectively. The dividing segment appears once in each perimeter and cancels. Equal perimeters therefore give

\begin{aligned} 3+4t+6t &=9+4(1-t)\\ &\quad+6(1-t). \end{aligned}

Hence

20t=16,

so

t=\frac{4}{5}.

Each leg is divided in the ratio

t:(1-t)=\frac45:\frac15=4:1.

Thus, the correct answer is C.

1957 AMC 12 第 49 题

答案：C

难度评级：1830

小提示：

大提示：

解答：

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