Question 1164701

The number of sides of two polygons differ by 4 and the number of diagonals differ by 30. How many sides are there in the polygon with the greater number of sides?
<pre>Let number of sides/diagonals, be n, and L, respectively
Then number of sides and diagonals of the smaller polygon are: n - 4, and L - 30, respectively
Formula for number of diagonals of a polygon: {{{n(n - 3)/2}}}
Formula for number of diagonals of larger polygon: {{{matrix(1,3, n(n - 3)/2, "=", L)}}} 
                                                   {{{matrix(1,3, n^2 - 3n, "=", 2L)}}} ----- eq (1)

Formula for number of diagonals of smaller polygon: {{{matrix(2,3, (n - 4)(n - 3 - 4)/2, "=", L - 30, (n^2 - 11n + 28)/2, "=", L - 30)}}}
                                                    {{{matrix(1,3, n^2 - 11n + 28, "=", 2L - 60)}}} ----- eq (ii) 
n<sup>2</sup> -  3n      = 2L -------- eq (1)
n<sup>2</sup> - 11n + 28 = 2L - 60 --- eq (ii) 
      8n - 28 = 60 ------ Subtracting eq (ii) from eq (i)
8n = 60 + 28
8n = 88
Number of sides of LARGER polygon, or {{{highlight_green(matrix(1,5, n, "=", 88/8, "=", 11))}}}</pre>