Problem 13 · 1998 AJHSME
Hard
Geometry & Measurement
area-ratiosymmetry
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Answer: C — 1/8.
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Hint 1 of 2
Don't measure the tilted square directly — relate it to easy pieces of the BIG square. The lines from the top corners meet at the center, carving the square into four matching triangles.
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Hint 2 of 2
Focus on the bottom quarter-triangle. What fraction of THAT triangle does the shaded square fill? Then 'fraction of a quarter' gives the fraction of the whole.
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Approach: compare the shaded square to one quarter of the big square
- Lines run from the two top corners down to the center of the square, splitting it into four equal triangles (top, bottom, left, right) — each is one quarter of the whole.
- The shaded tilted square sits inside the bottom quarter-triangle and covers exactly half of it (the extra vertical line marks the split).
- So the shaded area = ½ of ¼ = 1/8 of the large square.
- Why this transfers: for a tilted shape inside a square 'drawn to scale,' chop the big square into equal slices you trust, then express the target as a fraction of one slice — far safer than guessing lengths off a diagonal.
Another way — coordinates (pin the corners):
- Put the big square on a grid with corners (0,0), (4,0), (4,4), (0,4) — area 16. The shaded square's corners land at the center (2,2), the base midpoint (2,0), and the two points (1,1) and (3,1) where the corner-lines cross.
- That tilted square has diagonals of length 2 (from (2,0) to (2,2)) and 2 (from (1,1) to (3,1)); a square's area is ½·(diagonal)·(diagonal) = ½·2·2 = 2.
- Ratio = 2/16 = 1/8, confirming the slice argument.
Mark:
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