There is a relationship between the number of vertices in a

There is a relationship between the number of vertices in a polygon and the number of triangles in any triangulation of that polygon. State this relationship and prove it by induction.

Solution

the relationship is as follows

the number of triangles is n-2 , where n is the sides of polygon

for example if n=3 , that is our polygon is triangle then number of triangle by this formula is 1

if n=4 and we draw a non intersecting diagonal then number of triangle is 2

to prove by induction

LetP(n) be the statement that number of triangles is n-2

therefore for n=3 , i.e base case it is true , since in a triangle we have only one triangle

let P(n) be true for n >3

therefore we have to prove for n = n+1

draw a diagonal from a to b which cuts the polygon into two parts P1 and P2 where each of these polygons have n1 and n2 vertices respectively.

here n1+n2 = n+2 since we have a,b in both polgons as common

the inductive hypothesis says there are n1-2 and n2-2 number of triangles in P1 and P2

hence P has

n1-2 + n2-2

=> (n1+n2) -4

=>n+2 -4

=> n-2

hence true for the hypothesis also