### A polygon has 1325 diagonals. How many vertices does it have?

A polygon has 1325 diagonals. How many vertices does it have?

The formula to find number of diagonals (D) given total number of vertices or sides (N) is

N * (N - 3)

D = -----------

2

Using the formula, we get
1325 * 2 = N * (N - 3)
N2 - 3N - 2650 = 0

Solving the quadratic equation, we get N = 53 or -50

It is obvious that answer is 53 as number of vertices can not be negative.

Alternatively, you can derive the formula as triange has 0 diagonals, quadrangel has 2, pentagon has 5, hexagon has 9 and so on......

Hence the series is 0, 0, 0, 2, 5, 9, 14, ........ (as diagram with 1,2 or 3 vertices will have 0 diagonals).

Using the series one can arrive to the formula given above.

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