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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.

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.

D=

where:

D=diagonals

n=number of sides

Now substitute:

90=

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 =)