Problem 9 · 1988 AJHSME
Medium
Geometry & Measurement
distance-on-gridisosceles-check
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Answer: D — 4.
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Hint 1 of 2
Isosceles just means two sides are equal. On a grid you don't need the actual lengths — for each slanted side, the box it spans gives a 'right + up' pair (a, b). Two sides match exactly when their (a, b) boxes match (in either order).
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Hint 2 of 2
Skip the square roots: compare a² + b² for each side instead of the length itself. If two sides give the same a² + b², they're equal — and the triangle is isosceles.
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Approach: compare squared side lengths from the grid
- For each triangle, read every side as a 'right a, up b' move on the grid, and compute a² + b² (this is the side's length squared — no square roots needed). A triangle is isosceles the moment two of its three sides give the same value.
- Checking all five this way, four of them have a matching pair of sides; only one comes out with three different values (scalene). So 4 are isosceles.
- Why this transfers: lengths on a grid almost always come out as ugly square roots. Comparing a² + b² keeps everything as whole numbers, so you can decide 'equal or not' by eye without ever taking a root.
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