Question

Conjecture and Prove: A convex polygon is a figure drawn in the plane consisting of edges (straight line segments) and vertices. Edges can only meet at vertices, each vertex has exactly two edges that meet there, and each of the interior angles measures less than 180°. A diagonal of a polygon is a line segment that connects two non-adjacent vertices (adjacent vertices share an edge between them). Explore: how many diagonals convex polygons have and write a conjecture about the number of diagonals in an n-sided polygon: e number ses have and how many diac Conjecture: The number of unique diagonals that can be drawn in a convex, n-sided polygon is Prove your conjecture via induction (Note that n>3).

how can I do this problem? thank you for the help it's much needed

Answer 1

hou n(n-3) Conjecture- A convex polygon with n-sides Pin): 1" diagonals, for all nys. Potove :- 0 For n=4, we have a quad rilaterial, we know every quadrilateral has 2 diagonals. And 414-31-2 Thus, P(4) is true. 0 Let us assume, P(m) is tsure, for my? i.e. a m-sided convex polygon has mcm-3) diagonals, 0 Let us consider a (m+i) sided convene polygon. So, it has m+ vertex. Let one of them be u. So, If we delete this vertex we have m number of vertices. which will form m-sided polygon. which has mim-3) diagonals ( By Induction hypothesis) v Now, when we add V, These diogonals will still be diagonals of the neue M+ sided convex polygon. Also, v will have m-2 non adjacent vertices which will give m-2 diagonals. Also, if, vi & V2 are the adjacent vertices of v, The side viva in m-sided polygon will be a diagonal of the newe (m+1) -sided polygon. So, Total diagonals, mim-3) +(m-2)+1

2 (m2 - 3m) +20m -2) +2 2 m²-3m +2m - 4 + 2, ma- 1 m²m-2 2 m2m+m-2, (m-2)(m+1) - 2 2 (m +1) ( 01 -3) 2 . Thus, P(m+1) is also true, when, P(m) is true, & my P(n) is true by Induction, any3 & nEN.
If you have any problem while understanding any part,Leave a comment.I will try my best to help