Competition · AMC preparation · step 4 of 4

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 1/2s², so the three outer areas are exactly half the three squared sides — multiply Pythagoras by 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

Find the area W

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

W = 1/2 · 3 · 4 = 6

2STEP 2

Find the areas X, Y, Z

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 = 1/2 · 3² = 4.5, Y = 1/2 · 4² = 8, Z = 1/2 · 5² = 12.5

3STEP 3

Spot the Pythagorean pattern

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 → 1/2 · 3² + 1/2 · 4² = 1/2 · 5² → X + Y = Z

Grade 8 "apply the Pythagorean theorem" — three similar shapes on the sides of a right triangle always satisfy outer_a + outer_b = outer_c, because all three are the same fraction of the squares on the sides.

The outer triangle built on the long side covers the same area as the two outer triangles built on the two shorter sides combined.

▸ Why?

Each outer triangle is a right triangle whose two equal legs are both as long as the side it shares, so its area is half of that side times itself — exactly half the square on that side, giving 1/2 · 3², 1/2 · 4², and 1/2 · 5².

▸ Why?

Two copies of that right triangle, one flipped against the other, fit together into the full square on that side, so a single triangle is half of the square.

▸ Why?

Flipping one copy to lay it against the other does not change its size, so the two pieces are congruent halves of the square.

Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

▸ Why?

The two triangles fill the square with no gaps or overlaps, so their two areas add back to the whole square.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

With every area now written as half a square, the two smaller areas add up to the largest exactly when 3² + 4² = 5².

▸ Why?

Taking half of each square and then adding gives the same result as adding the squares first and then taking half, so 1/2 · 3² + 1/2 · 4² = 1/2(3² + 4²).

Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.

▸ Why?

The inner triangle has legs 3 and 4 meeting at a right angle with hypotenuse 5, so the squares on the two legs add up to the square on the hypotenuse, 3² + 4² = 5².

Pythagorean theoremIn a right triangle the two legs' squares add up to the hypotenuse's square (a²+b²=c²).

8.G.B.7Create A Physical Representation

4STEP 4

Match against the choices

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.

Find the area W
Find the areas X, Y, Z
Spot the Pythagorean pattern
Match against the choices

A parent dashboard for the family lives at sensimlab.com.

Find the area W

Find the areas X, Y, Z

Spot the Pythagorean pattern

Match against the choices

More like this