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The total number of interior angles in two regular polygons is 17, and the total number of diagonals is 53. How many sides does each polygon have? Show work and explain.

Think of the number of diagonals for an n-gon.

For each point p, there are n-3 points that form diagonals (n less p itself and the two adjacent vertices)

Going around for all n points, each diagonal is counted twice. So, the dumber of diagonals d(n) for a n n-gon is

d(n) = n(n-3)/2

The number of interior angles is just n.

So, for two polygons, of m and n vertices,

m + n = 17
m(m-3)/2 + n(n-3)/2 = 53

substitute m = 17-n

(17-n)(14-n)/2 + n(n-3)/2 = 53

You have a 6-gon and an 11-gon

Akeem is putting a fence around his square garden. If the area of the garden is 144 square meters, how many meters of fencing will he need?

11222

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