AMC 10 · 2006 · #6

Grade 6 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Rectilinear Frame / Box Border
area-rectanglesspatial-visualization convert-to-algebra ↑ Prerequisites: area-rectangles
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
A rectangle is cut along a staircase line into two identical hexagons that slide together into a square. Find the length of the short flat piece of the cut.

Pick an answer.

(A)
6
(B)
7
(C)
8
(D)
9
(E)
10
How to solve
Strategy Visualize Spatial Relationships

The heart of the problem is imagining the two hexagons sliding together into a square, so Tool #17 (Visualize Spatial Relationships) leads: picture the pieces moving. Tool #1 (Draw a Diagram) supports it — sketch the finished square to see where y lands. Tool #6 (Guess and Check) handles one small arithmetic step: find the number that, times itself, gives the square's area. The chain is short: area is preserved, so the square's side is fixed, and y is a simple fraction of that side.

1STEP 1

Area does not change

Sliding without overlap keeps the area unchanged.

Area = 8 × 18 = 144
2STEP 2

Find the square's side

So the square's side is 12.

s × s = 144 → s = 12 (12 × 12 = 144)
3STEP 3

Locate y in the square

Identical pieces meet at the middle of a side, so the answer is 6, choice (A).

y = 12/2 = 6 → (A)
Answer
6
Check the numbers against the picture: a side of 12 is a believable square built from an 8-by-18 block, since 12 sits between 8 and 18. And y = 6 is one third of the 18-length edge, which matches the cut starting well before the middle. The value 6 is the smallest choice offered and lands cleanly on a whole number, with no leftover gaps in the square.
💡Key takeaway

Cutting and rearranging keeps the area the same, so find the square's side first, then read off where the marked length lands.

  • Area does not change
  • Find the square's side
  • Locate y in the square

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