2009 AMC 10B Problem 4

Everything depends on the size of the triangles, so find their legs first.

1. In the picture, the bottom edge of the yard is the long side of the trapezoid, so the yard is $25$ meters long. The short side of the trapezoid, $15$, is the part of the top edge between the two triangles.
2. The top edge is also $25$ long, so the two triangles together take up $25 - 15 = 10$ meters of it. They are the same size, so each takes $5$ meters.
3. Each triangle is an isosceles right triangle, so its two legs are equal. One leg lies along the top edge ($5$ meters), so the other leg, along the short side of the yard, is also $5$ meters. That means the yard is $5$ meters wide.
4. Area of one triangle: $\frac12 \times 5 \times 5 = 12.5$. Both triangles: $25$.
5. Area of the yard: $25 \times 5 = 125$.
6. Fraction: $\frac{25}{125} = \frac15$.

The answer is $\boxed{\textbf{(C)}\ \frac15}$.

"Isosceles right triangle" means the two legs are equal, so knowing one leg gives you the other. The difference between the two parallel sides of the trapezoid is exactly what the triangles cover, and the triangle's leg is also the yard's width. One number, $5$, unlocks every area.

Find the trapezoid's area instead and subtract. Its parallel sides are $15$ and $25$ and its height is the yard's width, $5$, so its area is $\frac{15 + 25}{2} \times 5 = 100$. That leaves $125 - 100 = 25$ for the flower beds, and $\frac{25}{125} = \frac15$.