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2006 AMC 8

Problem 5

Problem 5 · 2006 AMC 8 Easy
Geometry & Measurement midpoint-square

Points A, B, C and D are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square?

Figure for AMC 8 2006 Problem 5
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Answer: D — 30.
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Hint 1 of 2
The midpoint lines cut off four corner triangles. Picture folding each triangle inward along the slanted line — where do the four corners land?
Still stuck? Show hint 2 →
Hint 2 of 2
Connecting the midpoints of any square always makes a tilted square with exactly HALF the area. Knowing this fact lets you skip all the computation.
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Approach: the four corner triangles fold in to fill the inner square
  1. The four slanted lines slice off four right triangles at the corners. Fold each one inward along its slanted edge.
  2. The four triangles exactly tile the inner diamond — so the inner square is made of half the big square's area, the other half being the (folded-out) triangles.
  3. Inner area = 60 ÷ 2 = 30.
  4. You'll see it again: the midpoint square of ANY square (or even any quadrilateral — the "Varignon" idea) has half the area. Remember the fact and these become instant.
Another way — side length from the half-diagonal:
  1. Let the big square have side s, so s2 = 60. Each midpoint sits halfway along a side.
  2. The inner square's side is the hypotenuse of a right triangle with legs s/2 and s/2, so (inner side)2 = (s/2)2 + (s/2)2 = s2/2.
  3. Inner area = (inner side)2 = s2/2 = 60/2 = 30 — the algebra confirms the half-area fact.
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