Question

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

Answer 1

Please check the answer if it is correct thank you..

Answer 2

Answer:

455 triangles are there on he vertices of a polygon.

Step-by-step explanation:

Let us consider:

X= no(vertices) of the polygon.

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

xd = p(p-3)/2

According to the question, we have

xd=90 from the question;

p(p-3)/2 = 90

=>p(p-3)=180;

=>p2-3p-180=0

=>p2-15p+12p-180=0

=>p(p-15)+12(p-15)=0;

=>(p+12)(p-15)=0;

=>p+12=0; p+15=0;

Therefore p=-12 and 15

N(vertices) should be positive, so p = 15.

To find the number of triangles on the vertices of the polygon is

Nt=15C3= $\frac{15 !}{3 !(15-3) !}=\frac{15 !}{3 ! 12 !}=\frac{15 \times 14 \times 13 \times 12 !}{3 \times 2 \times 1 \times 12 !}\frac{15 !}{3 !(15-3) !}$  =5×7×13=455

So the number of triangles is 455.