AMC 10 · 2006 · #7

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 ABCD measuring 8 by 18 is cut along a staircase line into two identical hexagons. Those two hexagons can be slid around (no flipping needed, no overlap) so they fit together into a perfect square. In the picture, y is the length of the short flat piece of the cut along the top-left edge (and the matching one on the bottom-right). Find y.

Pick an answer.

(A)
$\ 6$
(B)
$\ 7$
(C)
$\ 8$
(D)
$\ 9$
(E)
$\ 10$

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

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

Cutting and sliding gains or loses no area, so the square matches the rectangle: 8 × 18 = 144.

Area = 8 × 18 = 144
2STEP 2

Find the square's side

Which number times itself makes 144? Testing gives 12 × 12 = 144, so each side of the square is 12.

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

Locate y in the square

In the 12 × 12 square the y piece runs from a corner to the middle of a side, so y = 12 ÷ 2 = 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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