Problem 15 · 1997 AJHSME
Hard
Geometry & Measurement
pythagoreanarea-ratio
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Answer: B — 5/9.
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Hint 1 of 2
Trisection means clean thirds, so set the big side to 3 — then each tilted inner side is the slanted edge of a right triangle that goes 2 along and 1 up.
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Hint 2 of 2
For a tilted square, find its side via the Pythagorean theorem on the corner triangle; for an area RATIO you only ever need side², so the √ never has to be evaluated.
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Approach: Pythagoras gives side², which is the area
- Let the big square's side be 3 (trisection points fall at 1 and 2). Each side of the inner square is the hypotenuse of a right triangle with legs 2 and 1.
- By the Pythagorean theorem the inner side² = 2² + 1² = 5. (No need to take the square root!)
- Area ratio = inner side² / big side² = 5 / 3² = 5/9.
- Key shortcut: a square's area IS its side², so for area problems stop at side² = 5 — chasing √5 just invites it to be squared right back.
Another way — subtract the four corner triangles:
- The inner square is the big 3×3 square with four right triangles snipped off the corners, each with legs 2 and 1.
- Each corner triangle has area ½ · 2 · 1 = 1, so four of them total 4. Inner area = 9 − 4 = 5.
- Ratio = 5/9 = 5/9 — same answer with no Pythagoras at all, just area bookkeeping.
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