2009 AMC 10B 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?

\dfrac18

\dfrac16

\dfrac15

\dfrac14

\dfrac13

Answer: C

Concepts:triangle area trapezoid area ratio

Difficulty rating: 960

Small hint:

Each triangle’s legs are half of

25-15

Big hint:

Compare the total triangle area to the full rectangle of dimensions

25\times5

Solution:

The two parallel sides differ by

25-15=10,

split evenly between the two triangles, so each isosceles right triangle has legs of length

5

and area

\dfrac12\cdot5^2=\dfrac{25}{2}.

Together the beds cover

25

square meters. The rectangle has length

25

and width

5,

so area

125.

The fraction is

\dfrac{25}{125}=\dfrac15.

Thus, the correct answer is C.