AMC 8 · 2003 · #6

Grade 8 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

pythagorean-theoremperfect-squaresarea-trianglesspatial-visualization identify-subproblemspattern-recognition ↑ Prerequisites: perfect-squaresarea-trianglesmulti-digit-arithmetic

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A figure shows three squares meeting at the vertices of an interior triangle. The squares have areas 25, 144, and 169. Find the area of the interior triangle.

Pick an answer.

(A)

(B)

(C)

(D)

300

(E)

1800

AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Break Into Subproblems

The figure pairs each square with one side of the triangle, so Tool #7 (Break Into Subproblems) splits the problem into three clean parts: (1) turn each square's area into a side length, (2) check what kind of triangle has sides 5, 12, 13, (3) compute its area. Tool #10 (Use a Related Problem) is the recognition that 5-12-13 is a famous Pythagorean triple, so the converse of the Pythagorean theorem from a related problem makes step (3) easy — the triangle is right-angled, and its legs are the base and height.

1STEP 1

Take the square root of each area to get its square's side: 5, 12, and 13.

√(25) = 5, √(144) = 12, √(169) = 13

2STEP 2

Each triangle side is a shared square side, so the triangle has sides 5, 12, 13.

triangle sides = 5, 12, 13

3STEP 3

Since 5² + 12² = 169 = 13², the converse of the Pythagorean theorem makes it a right triangle with legs 5 and 12.

5² + 12² = 25 + 144 = 169 = 13² → right triangle with legs 5, 12

4STEP 4

The legs are the base and height, so the area is half their product: $\frac{1}{2}$ · 5 · 12 = 30.

Area =

\frac{1}{2}

· 5 · 12 = 30 → (B)

Answer

Cross-check using the answer choices. (A) 13 matches a side length, not an area — a trap. (E) 1800 and (D) 300 are far too big: the right triangle sits inside the 12 × 12 square, so its area cannot exceed 144. (C) 60 = 5 · 12 forgets the factor of

\frac{1}{2}

. Only (B) 30 is consistent with the bound and with

\frac{1}{2}

· 5 · 12, confirming the answer.

💡Key takeaway

Each square's area gives you a side of the triangle. Spot the 5-12-13 right triangle and the area is just $\frac{1}{2}$ · 5 · 12 = 30.

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