Question 483998: In a regular pentagon, lines are drawn so that every possible pair of vertices are connected. How many triangles are in the resulting figure?
choose the answer and explain the reason to choose your response
35
30
20
25
Found 2 solutions by richard1234, chessace:
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Counting them manually is probably the easiest way to do so. Once you draw the pentagon and all its diagonals, you can find five triangles, each one having three consecutive vertices on the pentagon (e.g. V1V2V3, V2V3V4 if we label the vertices V1,...,V5). You can count six triangles in each of them, so 30 triangles. There are also five additional triangles determined by non-adjacent vertices (e.g. V1V2V4, etc.) so the answer is 35.
Answer by chessace(471)  (Show Source):
You can put this solution on YOUR website! The major problem with this is to avoid counting a triangle twice.
Each edge has a small triangle with no cross lines: 5
Each vertex has a small triange with no cross lines: 5
Each edge has 2 triangles with 1 cross line (its small triagle merged with one of the small triagles of its 2 vertices): 10
Each vertex (using its 2 original edges) has a triangle with 2 cross lines: 5
Each original edge to the opposite vertex has a triangle with 5 little cross lines: 5
Total = 30.
The previously posted solution counted the small triangle by an edge twice, hence over by 5.
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