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2009 AMC 10B Problem 4

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2009 AMC 10B Problem 4AMC 10 Step by Step
Name:                             Date: September 13, 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 1515 and 2525 meters. What fraction of the yard is occupied by the flower beds?
figure

  1. A)

    18\frac {1}{8}

  2. B)

    16\frac {1}{6}

  3. C)

    15\frac {1}{5}

  4. D)

    14\frac {1}{4}

  5. E)

    13\frac {1}{3}

Answer

C

Key insight

The two triangle bases fill the 25 - 15 = 10 meters left over on the long side, so each leg is 5, which is also the yard's width.

Solution

The long side of the yard is the trapezoid's long base, 2525. The short base, 1515, sits along the top edge between the two triangles, so the two triangles together take up 25−15=1025 - 15 = 10 meters of the top edge, or 55 each.

Each flower bed is an isosceles right triangle with one leg along the top edge, so both legs are 55. The other leg runs along the side of the yard, which means the yard is 55 meters wide.

Areas: each triangle is 12⋅5⋅5=252\frac12 \cdot 5 \cdot 5 = \frac{25}{2}, so the beds total 2525. The yard is 25⋅5=12525 \cdot 5 = 125. The fraction is 25125=15\frac{25}{125} = \frac15.

The answer is (C) 15\boxed{\textbf{(C)}\ \frac15}.

Why this works

"Isosceles right triangle" is a strong hint: one measured leg gives everything else. The difference between the trapezoid's bases is shared by the two congruent triangles, and the leg length doubles as the rectangle's width, so a single number unlocks all areas.

Alternative approach

The trapezoid has area 15+252⋅5=100\frac{15 + 25}{2} \cdot 5 = 100 out of 125125, leaving 2525 for the beds: again 15\frac15.

The trap

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

Common mistakes

  • Using 10 as the leg of each triangle instead of splitting the 10 meters between the two congruent triangles.
  • Computing the fraction of the trapezoid rather than of the whole yard, giving 14\frac14.

Techniques

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