2009 AMC 12B Problem 4

Question

A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths

15

and

25

meters. What fraction of the yard is occupied by the flower beds?

\dfrac{1}{8}

\dfrac{1}{6}

\dfrac{1}{5}

\dfrac{1}{4}

\dfrac{1}{3}

Answer: C

Concepts:area ratio triangle area trapezoid

Difficulty rating: 1100

Small hint:

The two triangle legs equal half the difference of the parallel sides

Big hint:

Each isosceles right triangle has legs

\dfrac{25 - 15}{2} = 5

Solution:

The parallel sides differ by

25 - 15 = 10,

so each triangle has legs

\dfrac{10}{2} = 5

and area

\dfrac{1}{2} \cdot 5^2 = \dfrac{25}{2}.

The two beds total

25.

The rectangle measures

25

by

5,

so its area is

125,

and the fraction occupied is

\dfrac{25}{125} = \dfrac{1}{5}.

Thus, the correct answer is C.