SOLUTION: 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?

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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?
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
the number of diagonals in a polygon is equal to n * (n-3) / 2.

if the number of sides of the larger polygon is 4 more than the number of sides of the smaller polygon, then:

the number of sides of the larger polygon is equal to n + 4, while the number of sides of the smaller polygon is equal to n.

the number of diagonals of the smaller polygon is equal to n * (n-1) / 2.

the number of diagonals of the larger polygon is equal to (n + 4) * (n + 4 - 3) / 2 which is equal to (n + 4) * (n + 1) / 2.

if 30 is the result when the number of diagonals of the smaller polygon is subtracted from the number of diagonals of the larger polygon, then your formula becomes:

(n + 4) * (n + 1) / 2 - n * (n - 3) / 2 = 30.

multiply both sides of this equation by 2 to get:

(n + 4) * (n + 1) - n * (n - 3) = 60.

simplify this to get n^2 + n + 4n + 4 - (n^2 - 3n) = 60

simplify this further to get n^2 + n + 4n + 4 - n^2 + 3n = 60

combine like terms to get:

8n + 4 = 60.

subtract 4 from both sides of the equation and simplify to get:

8n = 56

solve for n to get n = 56 / 8 = 7

that's the number of sides of the smaller polygon.

the number of the sides of the larger polygon is 7 + 4 = 11

to confirm, do the following:

when n = 11, the number of diagonals is equal to 11 * (11 - 3) / 2 = 11 * 8 / 2 = 11 * 4 = 44

when n = 7, the number of diagonals is equal to 7 * (7 - 3) / 2 = 7 * 4 / 2 = 7 * 2 = 14

the difference in number of diagonals is 44 minus 14 = 30

this confirms the solution is correct.

your solution is that the number of sides of the larger polygon is 11.

Answer by MathTherapy(10549) (Show Source):
You can put this solution on YOUR website!

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?

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: Formula for number of diagonals of larger polygon: ----- eq (1) Formula for number of diagonals of smaller polygon: ----- eq (ii) n² - 3n = 2L -------- eq (1) n² - 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