# In a convex polygon, the number of diagonals outgoing from the vertex is 15. Find the number of all diagonals of this polygon.

In a convex polygon, the number of diagonals drawn from one vertex is 3 less than the number of vertices of this polygon.

Let’s find the number of vertices of this polygon:

d₁ = n – 3;

n = d + 3;

n = 15 + 3 = 18.

The number of all possible drawn diagonals in a polygon is found by the formula:

d = (n² – 3 * n) / 2,

where d is the number of possible different diagonals, n is the number of vertices of the polygon.

We found out that the polygon has 18 vertices. Substitute this value into the formula and find the number of all possible diagonals of the eighteen-gon:

d = (18² – 3 * 18) / 2 = (324 – 54) / 2 = 270/2 = 135.

Answer: the number of all diagonals of a polygon in which 15 diagonals can be drawn from one vertex is 135.

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