Question 96363
Although it is true that from any vertex of a regular octagon lines can be drawn to 7 other 
vertices, those lines are NOT diagonals.
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In a regular octagon, from a vertex you can only draw 5 diagonals. A line from a vertex to the next 
adjacent vertex on either side is not a diagonal.
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Let's use something you are familiar with. If you are given a square (which has four 
sides and is a quadrilateral), how many diagonals can you draw from one corner? The answer 
is 1 diagonal from that corner, not 4 - 1 = 3. From one vertex in a square the only
diagonal you can draw is from that vertex to the vertex in the opposite corner.
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In fact, for any regular geometric figure the answer to this question is N - 3 where N is 
the number of sides of the given figure.
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So, from one vertex in:
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a regular triangle you can draw  3 - 3 = 0 diagonals
a regular quadrilateral you can draw 4 - 3 = 1 diagonal
a regular pentagon you can draw 5 - 3 = 2 diagonals
a regular hexagon you can draw 6 - 3 = 3 diagonals
a regular heptagon you can draw 7 - 3 = 4 diagonals
a regular octagon you can 8 - 3 = 5 diagonals
and so on
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Hope this helps you to see why the answer to your problem is 5 diagonals, not 7.
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