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2015-11-02T17:14:11+08:00

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X + y = 13
y = 13 - x

Representation:
x = total number of sides of first polygon
13 - x = total number of sides of second polygon.

Number of diagonals in a polygon:

 \frac{n(n-3)}{2}
where n = number of sides of a polygon 

Number of diagonals in first polygon with x sides.
Substitute x for n:
=  x (x-3)  
       2
= x² - 3x
       2

Number of diagonals in second polygon with 13-x sides.  
Substitute 13 - x for n:
=  13 - x (13 - x - 3)  
            2
= 13 - x (10 - x)  
           2
= 130 -13x - 10x + x²
           2
= x² - 23x + 130
          2

The sum of diagonals of the two polygons is 25.

x
² - 3x + x² - 23x + 130 = 25
   2                  2

x² + x² -3x - 23x + 130 = 25
          2

2 (2x² - 26x + 130 = 25) 2
          2

2x² - 26x + 130 = 50
2x² - 26x + 130 - 50 = 0
2x² - 26x + 80 = 0

Factor out 2 (GCF)
2 (x² - 13x + 40) = 0

Factor the quadratic equation, then solve for the roots (x):
x² - 13x + 40 = 0
(x - 8) (x - 5) = 0
x - 8 = 0                   x - 5 = 0
x = 8                        x = 5

The number of sides of each polygon:
First polygon:x              ⇒  x = 8 sides (octagon)
Second polygon: 13 - x  ⇒  13 - 8 = 5 sides (pentagon)

To check if 8 and 5 sides are correct:
The sum of the sides of the two polygons is 13
8 + 5 = 13
13 = 13

The sum of the diagonals of the two polygons is 25:
Diagonals of first polygon with 8 sides, where n = sides:
n (n-3) = 8 (8-3) = 8(5) = 40  = 20 diagonals
   2           2           2       2

Diagonals of the second polygon with 5 sides, where n = sides:
n(n-3)  = 5 (5-3)  =  5(2) =  10   =  5  diagonals
  2            2            2         2

Add the diagonals:
20 + 5 = 25
25 = 25

FINAL ANSWER:  The number of sides of the two polygons are 8 and 5 sides.

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