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2015-12-12T09:34:48+08:00

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The name of a regular polygon can be name as n-gon where where n is the number of sides of a polygon.

In order to find the name of a regular polygon, we have to get the number of sides of a polygon with the given number of diagonals by using the formula
no. of diagonals = n(n-3) / 2  where n is the no. of sides
90 = n(n-3) / 2
90 = (n^2 - 3n) / 2
90×2 = n^2 - 3n
180 = n^2 - 3n
0 = n^2 - 3n - 180
0 = (n - 15)(n + 12)
n - 15 = 0              n + 12 = 0
n = 15                    n = -12

We have two values of n which are 15 and -12. But we will not consider n = -12 because there is no possibility to have a negative number of sides of a polygon.
So we have 15 number of sides of a polygon.

Therefore the name of a regular polygon that has 90 diagonals is 15-gon or pendedecagon.
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2015-12-12T16:40:10+08:00
There is a formula for finding the number of sides of a polygon given diagonals, which is:
D=\frac{n(n-3)}{2}
where:
D=diagonals
n=number of sides

Now substitute:
90=\frac{n(n-3)}{2}
Multiply both sides with 2
180=n(n-3)
Distribute n
180=n²-3n
Equate it to Ax²+By+C=0
n²-3n-180=0
Factor:
(n-15)(n+12)=0
n-15=0   n+12=0
n=15      n=-12
Since there are no negative sides, then we consider n=15
the name of the polygon that has 90 diagonals is pentadecagon
Hope this helps =)
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