AMC 8 · 2011 · #20

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

area-trianglesarea-rectanglespythagorean-theorem identify-subproblemsarea-difference ↑ Prerequisites: pythagorean-theoremarea-trianglesarea-rectangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Trapezoid ABCD has parallel sides AB = 50 (top) and CD (bottom), slant legs AD = 15 and BC = 20, and a height of 12 between the parallel sides. Find its area.

Pick an answer.

(A)

600

(B)

650

(C)

700

(D)

750

(E)

800

AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The figure is given, but the key move — dropping altitudes from A and B down to CD — has to be added by hand. That is Tool #1 (Draw a Diagram): augment the picture so the structure is visible. Once those two altitudes are drawn, Tool #7 (Identify Subproblems) takes over: the trapezoid splits into a left right triangle (legs 12 and a, hypotenuse 15), a middle rectangle of width AB = 50, and a right right triangle (legs 12 and b, hypotenuse 20). Each piece is easy on its own, and CD = a + 50 + b falls out. No algebra (Tool #13) is needed.

1STEP 1

Drop altitudes from A and B onto CD, splitting the trapezoid into rectangle ABYX of width 50 plus right triangles ADX and BCY.

AX = BY = 12; XY = AB = 50

2STEP 2

Pythagoras on right triangle ADX (legs 12 and a, hypotenuse AD = 15) gives horizontal piece a = 9.

a² + 12² = 15² → a² = 225 - 144 = 81 → a = 9

3STEP 3

Same on right triangle BCY (legs 12 and b, hypotenuse BC = 20) gives b = 16.

b² + 12² = 20² → b² = 400 - 144 = 256 → b = 16

4STEP 4

Add the three horizontal pieces to rebuild the bottom base: CD = 9 + 50 + 16 = 75.

CD = 9 + 50 + 16 = 75

5STEP 5

Plug bases 50 and 75 with height 12 into the trapezoid area formula to get area = 750.

Area =

\frac{1}{2}

(b₁ + b₂)h =

\frac{1}{2}

(50 + 75)(12) = 6 × 125 = 750 → (D)

Answer

750

Sanity check by adding the three pieces directly. Left triangle:

\frac{1}{2}

× 9 × 12 = 54. Middle rectangle: 50 × 12 = 600. Right triangle:

\frac{1}{2}

× 16 × 12 = 96. Total = 54 + 600 + 96 = 750, matching the formula answer (D). Magnitude check: the trapezoid is wider than 50 and taller than 0 but less than a 75 × 12 = 900 rectangle, and 750 sits comfortably in that window.

💡Key takeaway

Drop two altitudes and the trapezoid becomes a rectangle plus two right triangles — then Grade 8 Pythagorean theorem and the Grade 6 area formula finish it off.

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