Question

A polygon has 90 diagonals how many triangles are there on its vertices

Answer 1

The best way to solve this problem is to use the formula: An n sided polygon can have n*(n - 3)/2 diagonals 

n*(n - 3)/2 = 90
n*(n - 3) = 180

Substitute n from the answer choices. n = 15

Answer 2

Answer:  The number of triangles is 455.

Step-by-step explanation:  Given that a polygon has 90 diagonals. We are to find the number of triangles on the vertices of the polygon.

Let, 'n' be the number of vertices of the polygon.

We know that if 'n' is the number of vertices of a polygon, then the number of diagonals of the polygon is given by

$N_d=\dfrac{n(n-3)}{2}.$

According to the question, we have

$N_d=90\\\\\Rightarrow \dfrac{n(n-3)}{2}=90\\\\\Rightarrow n(n-3)=180\\\\\Rightarrow n^2-3n-180=0\\\\\Rightarrow n^2-15n+12n-180=0\\\\\Rightarrow n(n-15)+12(n-15)=0\\\\\Rightarrow (n-15)(n+12)=0\\\\\Rightarrow n-15=0,~~~~~~n+12=0\\\\\Rightarrow n=15, -12.$

Since the number of vertices cannot be negative, so n = 15.

Now, the number of triangles formed by 15 vertices of the polygon is given by the combination of 15 vertices taking 3 at a time.

Therefore, the total number of triangles is

$N_t=^{15}C_3=\dfrac{15!}{3!(15-3)!}=\dfrac{15!}{3!12!}=\dfrac{15\times 14\times 13\times 12!}{3\times 2\times 1\times 12!}=5\times 7\times 13=455.$

Thus, the number of triangles is 455.