AMC 8 · 2002 · #16

Grade 8 geometry-2d

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

pythagorean-theoremarea-trianglesperfect-squares identify-subproblemspattern-recognition ↑ Prerequisites: area-trianglespythagorean-theorem

📏 Short solution 💡 2 insights 📊 Diagram

Problem

A 3-4-5 right triangle has a right isosceles triangle built outward on each of its three sides. Call W the area of the original right triangle, and X, Y, Z the areas of the isosceles triangles on the sides of length 3, 4, 5 respectively. Which of the five area equations is always true?

Pick an answer.

(A)

X+Z=W+Y

(B)

W+X=Z

(C)

3X+4Y=5Z

(D)

$X+W=\frac{1}{2}(Y+Z)$

(E)

X+Y=Z

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Use a Related Problem

The setup — three shapes built on the sides of a right triangle, one per side — is the picture that appears in every proof of the Pythagorean theorem. Tool #10 (Use a Related Problem) tells us to lean on 3² + 4² = 5² instead of grinding through five answer choices. Each outer triangle on a side of length s has area $\frac{1}{2}$ s², so the three outer areas are exactly half the three squared sides — multiply Pythagoras by $\frac{1}{2}$ and the equation X+Y=Z pops out. Tool #7 (Break Into Subproblems) handles the bookkeeping: compute W, X, Y, Z one at a time, then test the choices.

1STEP 1

Inner 3-4-5 triangle: legs 3 and 4 form the right angle, so its area is W = 6.

W =

\frac{1}{2}

· 3 · 4 = 6

2STEP 2

Each outer 45-45-90 triangle has both legs equal to its shared side s, so area $\frac{1}{2}$ s²: X = 4.5, Y = 8, Z = 12.5.

X =

\frac{1}{2}

· 3² = 4.5, Y =

\frac{1}{2}

· 4² = 8, Z =

\frac{1}{2}

· 5² = 12.5

3STEP 3

The inner triangle is 3-4-5, so 3² + 4² = 5²; halve both sides and the outer areas satisfy X + Y = Z.

3² + 4² = 5² →

\frac{1}{2}

· 3² +

\frac{1}{2}

· 4² =

\frac{1}{2}

· 5² → X + Y = Z

4STEP 4

Substituting W=6, X=4.5, Y=8, Z=12.5, only choice (E) holds: X + Y = 12.5 = Z.

X+Y = 4.5 + 8 = 12.5 = Z → (E)

Answer

X+Y=Z

Two independent paths give the same answer. Direct computation: W=6, X=4.5, Y=8, Z=12.5, and only (E) checks out numerically. Structural argument: each outer area equals

\frac{1}{2}

of the square on its side, so the Pythagorean equation 3²+4²=5² scales by

\frac{1}{2}

to X+Y=Z for any right triangle, not just 3-4-5. The structural view also explains why W never appears in the correct equation — W is the inner triangle, not built on any side.

💡Key takeaway

Three triangles built on a 3-4-5 right triangle is just the Pythagorean picture in disguise — each area is half of a square on a side, so 3²+4²=5² becomes X+Y=Z.

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