2001 AMC 8 Problem 23
Attempt Problem 23 of the 2001 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2001 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
23.
Points and are vertices of an equilateral triangle, and points and are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices?
Answer: D
Small hint:
Classify triangles by side lengths
Big hint:
Use symmetry to avoid recounting congruent cases
Solution:
Using the side length of the small equilateral triangles as one unit, the possible triangle side-length patterns are:
equilateral with sides ; equilateral with sides ; small isosceles with sides ; and larger isosceles with sides .
These four patterns all occur in the figure, and symmetry shows every triangle formed by three of the six points matches one of them.
Thus there are noncongruent triangles.
Thus, D is the correct answer.
Problem 23 in other years
- 1985 AMC 8
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