Problem 20 · 2009 AMC 8
Hard
Geometry & Measurement
non-congruent-triangles
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
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Answer: D — 8.
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Hint 1 of 2
The 8 points form 2 rows of 4. Any triangle must use BOTH rows — three points on one line make no triangle — so it's always two points on one row and one point on the other.
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Hint 2 of 2
"Non-congruent" means count SHAPES, not positions: a triangle and its mirror image are the same shape. So organize by the horizontal gap between the two same-row points (1, 2, or 3) and how far the lone point sits from them, then drop mirror duplicates.
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Approach: classify by the pair's width and the lone point's offset, count shapes up to mirror
- Every triangle = two points on one row + one on the other (the rows are 1 unit apart). What fixes the SHAPE is two numbers: the gap g between the same-row pair, and the lone point's horizontal offset from them. Slide-and-flip copies are congruent, so count each distinct (g, offset) once.
- Gap g = 1 (adjacent pair, like columns 1–2): the lone point can sit directly under one end (offset 0), or 1, or 2 columns out to the side. The two sides are mirror images, so the distinct shapes come from offsets 0, 2, 3 measured from the pair — 3 shapes.
- Gap g = 2 (pair like columns 1–3): lone point under an end, under the middle, or one column beyond — 3 shapes.
- Gap g = 3 (pair at the far ends, columns 1–4): the lone point is at one of the two inner columns; those two choices are mirror images of each other, and being under an endpoint repeats a thinner shape — 2 shapes.
- Total distinct shapes: 3 + 3 + 2 = 8.
- Why this transfers: for "how many non-congruent figures," never list raw positions — pick the few measurements that pin down a shape (here a base length and an offset) and count those, folding mirror images together.
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