Problem 23 · 2001 AMC 8
Stretch
Geometry & Measurement
symmetrycareful-counting
Points R, S, and T are vertices of an equilateral triangle, and points X, Y, and Z are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices?
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Answer: D — 4.
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Hint 1 of 2
"Noncongruent" is the whole trap — you're counting distinct SHAPES, not positions. The figure's threefold symmetry means many triangles are rotated/reflected copies.
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Hint 2 of 2
Sort triangles by their ingredients: how many corners (R,S,T) versus midpoints (X,Y,Z) each one uses. Triangles with the same recipe come out congruent.
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Approach: count shapes up to congruence using symmetry
- Picking any 3 of the 6 points gives C(6,3) = 20 triangles — that's why 20 is offered as a tempting wrong answer. But the equilateral symmetry rotates and reflects most of them onto each other, so we count shape types, not all 20.
- Classify by makeup. 3 corners: only RST (big equilateral). 3 midpoints: only XYZ (small equilateral). 2 corners + 1 midpoint: e.g. R-T-Z, a 30-60-90 right triangle. 1 corner + 2 midpoints: e.g. R-Y-Z, an obtuse isosceles. (Three collinear points like R-X-T form no triangle.)
- That's exactly 4 distinct shapes. The reusable trick: when a figure is highly symmetric, group your choices by their structural "recipe" — copies under rotation/reflection collapse together.
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