# Diagonal

• Jan 7th 2008, 12:14 AM
vivian
Diagonal
Consider a convex n-gon such that no 3 diagonals intersect at a single point. Draw all the diagonals (i.e. connect every pair of vertices by a segment).
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a) How many intersections do the diagonals determine?

b) Into how many parts is the polygon divided by the diagonals?
• Jan 7th 2008, 02:04 AM
earboth
Quote:

Originally Posted by vivian
Consider a convex n-gon such that no 3 diagonals intersect at a single point. Draw all the diagonals (i.e. connect every pair of vertices by a segment).

a) How many intersections do the diagonals determine?

b) Into how many parts is the polygon divided by the diagonals?

Hello,

to a). from each vertex there are starting (n-3) diagonals.

The total number of diagonals is therefore: $\displaystyle \frac n2 \cdot (n-3)$

Each diagonal is intersected by (n-3) diagonals. Therefore the number of intersections is: $\displaystyle \frac n2 \cdot (n-3)^2$

to b) I'm not quite sure what you mean by parts :confused: