2001 AMC 8 Problems/Problem 23
Problem
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?
Solution
There are points in the figure, and
of them are needed to form a triangle, so there are
possible triples of
of the
points. However, some of these created congruent triangles, and some don't even make triangles at all.
Case 1: Triangles congruent to There is obviously only
of these:
itself.
Case 2: Triangles congruent to There are
of these:
and
.
Case 3: Triangles congruent to There are
of these:
and
.
Case 4: Triangle congruent to There are again
of these:
and
.
However, if we add these up, we accounted for only of the
possible triplets. We see that the remaining triplets don't even form triangles; they are
and
. Adding these
into the total yields for all of the possible triplets, so we see that there are only
possible non-congruent, non-degenerate triangles,
See Also
2001 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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