1955 AMC 12 Problem 31

Question

An equilateral triangle whose side is

2

is divided into a triangle and a trapezoid by a line drawn parallel to one of its sides. If the area of the trapezoid equals one-half of the area of the original triangle, the length of the median of the trapezoid is:

\dfrac{\sqrt6}{2}

\sqrt2

2+\sqrt2

\dfrac{2+\sqrt2}{2}

\dfrac{2\sqrt3-\sqrt6}{2}

Answer: D

Concepts:similar triangles trapezoid midsegment area ratio

Difficulty rating: 1630

Small hint:

The small triangle has half the original area, so determine its side using similarity

Big hint:

The trapezoid median is the average of its two parallel side lengths

Solution:

The small triangle also has half the original area. If its side parallel to the original base has length

b,

similarity gives

\left(\frac b2\right)^2=\frac12,

so

b=\sqrt2.

The parallel sides of the trapezoid have lengths

2

and

\sqrt2,

so its median has length

\frac{2+\sqrt2}{2}.

Thus, the correct answer is D.