1957 AMC 12 Problem 49

Question

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

Answer: C

Concepts:trapezoid ratio and proportion perimeter

Difficulty rating: 1830

Small hint:

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

Big hint:

Let the upper leg segments be

4t

and

6t

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

Solution:

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.