Answer: 90
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Work Shown:
n = number of vertices = 15
d = number of diagonals
[tex]d = \frac{n*(n-3)}{2}\\\\d = \frac{15*(15-3)}{2}\\\\d = \frac{15*12}{2}\\\\d = \frac{180}{2}\\\\d = 90\\[/tex]
So there are 90 diagonals for this 15-gon.
The numerator n(n-3) is from the fact that there are n vertices, and for each vertex, there are n-3 points to connect to that will form a diagonal. Why n-3? The 3 comes from the fact that you cannot select the current point, or either neighboring point (on either side). So we ignore 3 points when forming diagonals for any vertex. That's why n drops to n-3.
The 2 in the denominator is to correct for double-counting. A diagonal like AB is the same as BA. The order does not matter when it comes to naming a segment.
