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2009 AMC 10B Problem 4

2009 AMC 10B Problem 4AMC 10 Step by Step

Name: Date: October 28, 2026

Quadrilaterals & Polygon Areas · difficulty 1 of 5 · about 2 min · position P1-10

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?

A) $\frac {1}{8}$
B) $\frac {1}{6}$
C) $\frac {1}{5}$
D) $\frac {1}{4}$
E) $\frac {1}{3}$

Answer

Key insight

The two triangles use up 25 - 15 = 10 meters of the top edge, so each has legs of 5, which is also the yard's width.

Solution

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

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.
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.
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.
Area of one triangle: $\frac12 \times 5 \times 5 = 12.5$ . Both triangles: $25$ .
Area of the yard: $25 \times 5 = 125$ .
Fraction: $\frac{25}{125} = \frac15$ .

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

Why this works

"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.

Alternative approach

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 $\frac{25}{125} = \frac15$ .

The trap

Using 10 as the leg of each triangle, instead of splitting the 10 meters between the two triangles.

Concise solution

The long side of the yard is the trapezoid's long base, $25$ . The short base, $15$ , sits along the top edge between the two triangles, so the two triangles together take up $25 - 15 = 10$ meters of the top edge, or $5$ each.

Each flower bed is an isosceles right triangle with one leg along the top edge, so both legs are $5$ . The other leg runs along the side of the yard, which means the yard is $5$ meters wide.

Areas: each triangle is $\frac12 \cdot 5 \cdot 5 = \frac{25}{2}$ , so the beds total $25$ . The yard is $25 \cdot 5 = 125$ . The fraction is $\frac{25}{125} = \frac15$ .

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

Review in the AoPS books

Introduction to Geometry, Chapter 8: Quadrilaterals
Introduction to Geometry, Chapter 4: Perimeter and Area (§4.2)
Introduction to Geometry, Chapter 9: Polygons (§9.3)
Introduction to Geometry, Chapter 6: Right Triangles (§6.1, §6.2, §6.3)

Techniques

Cut the figure into known shapes (triangles, rectangles, sectors)