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.

1955 AMC 12 Problem 31

Answer: D

Difficulty rating: 1630

Small Hint:

Big Hint:

Solution:

Problem 31 in Other Years