AMC 10 · 2008 · #10

Grade 6 geometry-2d
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
area-trianglessimilar-figurespattern-recognition pattern-recognitionphysical-representation ↑ Prerequisites: area-triangles
📏 Medium solution 💡 2 insights
Problem
A square S₁ has area 16. Marking the midpoint of each side and joining those four points makes a smaller square S₂. Doing the exact same midpoint construction on S₂ makes an even smaller square S₃. Find the area of S₃.

Pick an answer.

(A)
$\ \frac{1}{2}$
(B)
$\ 1$
(C)
$\ 2$
(D)
$\ 3$
(E)
$\ 4$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

The problem repeats one move: join the midpoints, get a smaller square, do it again. Tool #5 (Look for a Pattern) says: figure out what one step does to the area, then that rule applies to every step. To find the rule, Tool #1 (Draw a Diagram) makes the midpoints and the tilted inner square visible, and Tool #7 (Identify Subproblems) splits the big square into the inner square plus four corner triangles so the area is easy to track. Once one step is understood, the pattern hands over S₃ with no new work.

1STEP 1

Draw the square and mark the midpoints

Area 16 means S₁ has side 4, so each half-side is 2, and the four midpoints are the corners of S₂.

side of S₁ = 4, each half-side = 2
2STEP 2

Cut off the four corner triangles

Each corner loses a right triangle with legs 2 and area 2, so the four corners remove 8 in all.

one triangle = 1/2(2)(2) = 2, four triangles = 8
3STEP 3

See that one step halves the area

So S₂ has area 16 - 8 = 8, exactly half of 16, and joining midpoints always leaves half the area.

[S₂] = 16 - 8 = 8 = 1/2 [S₁]
4STEP 4

Apply the halving pattern again to reach S3

The same construction halves again, so the areas run 16, 8, 4 and S₃ has area 4, choice (E).

[S₃] = 1/2 [S₂] = 1/2 × 8 = 4 → (E)
Answer
4
Two identical halvings turn 16 into 16 × 1/2 × 1/2 = 4, matching (E). The answer must be less than 8 (the area of S₂) but still a real square, and 4 fits; tiny values like 1/2 or 1 would need far more than two halvings of 16, so they are too small.
💡Key takeaway

Joining the midpoints of a square always leaves exactly half the area, so the areas go 16, then 8, then 4.

  • Draw the square and mark the midpoints
  • Cut off the four corner triangles
  • See that one step halves the area
  • Apply the halving pattern again to reach S3

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