44 or 45 Diagonals

What is the number of sides of a regular polygon for which the number of diagonals is 44 or 45?
Source: NCTM Mathematics Teacher 2006

SOLUTION
The number of diagonals of a convex polygon of n vertices (or sides) is given by n(n-3)/2. Please refer to post titled “Convex Polygon” posted on 6/18/2011 for details.
Case 1: 44 diagonals
n(n-3)/2=44
n(n-3)=88
Let’s try a few numbers
7(4)=28
8(5)=40
9(6)=54
10(7)=70
11(8)=88
The polygon has 11 sides.
Case 2: 45 diagonals
n(n-3)=90
Continuing where we left off above we get
12(9)=108
13(10)=130
\cdots
Clearly, there is no polygon that has 45 diagonals.

Answer: 11

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About mvtrinh

Retired high school math teacher.
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