A regular polygon with 'x' sides has 54 diagonals, while

Solution

Number of diagonals in a polygon with 'n' sides = n X {(n - 3)/2} So, 54 = x X {(x - 3)/2} Or, 108 = x² - 3x Or, x² - 3x - 108 = 0 Or, x² - 12x + 9x - 108 = 0 Or, x(x - 12) + 9(x - 12) = 0 Or, (x - 12)(x + 9) = 0 So, x = 12 or x = -9 Since, number of sides cannot be negative. So, x = 12 Similarly, 90 = y X {(y - 3)/2} Or, 180 = y² - 3y Or, y² - 3y - 180 = 0 Or, y² - 15y + 12y - 180 = 0 Or, y(y - 15) + 12(y - 15) = 0 Or, (y - 15)(y + 12) = 0 So, y = 15 or y = -12 Since, number of sides cannot be negative. So, y = 15 So, required sum = 12 + 15 = 27