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Sagot :
Let x and y be the polygons:
x + y = 15
First Polygon = x
Second Polygon: 15-x
x + y = 15
y = 15-x
Number of Diagonals in each polygon = n (n-3)
2
Where n = number of sides of regular polygon
Number of Diagonals for First Polygon, x:
= x(x-3)
2
Number of Diagonals for Second Polygon, 15-x:
= (15-x) (15-x-3) or (x-12)(x-15)
2 2
Add the diagonals of the two polygons. The sum is 36.
[tex]( \frac{x(x-3)}{2} )+( \frac{(x-12)(x-15)}{2} ) = 36[/tex]
x² - 15x + 90 = 36
x² - 15x + 90-36 = 0
x² - 15x + 54 = 0
Solve by factoring:
(x-9) (x-6) = 0
x - 9=0 x - 6 = 0
x = 9 x = 6
The two polygons have sides of 6 and 9:
Hexagon = 6 sides
Nonagon = 9 sides
To check:
The sum of sides of two polygons is 15
6 + 9 = 15
The diagonals:
Polygon with 6 sides = 6 (6-3)
2
= 6(3)
2
= 18/2
= 9 diagonals
Polygon with 9 sides = 9 (9-3)
2
= 9 (6)
2
= 54/2
= 27 diagonals
The sum of the number of diagonals:
9 + 27 = 36
x + y = 15
First Polygon = x
Second Polygon: 15-x
x + y = 15
y = 15-x
Number of Diagonals in each polygon = n (n-3)
2
Where n = number of sides of regular polygon
Number of Diagonals for First Polygon, x:
= x(x-3)
2
Number of Diagonals for Second Polygon, 15-x:
= (15-x) (15-x-3) or (x-12)(x-15)
2 2
Add the diagonals of the two polygons. The sum is 36.
[tex]( \frac{x(x-3)}{2} )+( \frac{(x-12)(x-15)}{2} ) = 36[/tex]
x² - 15x + 90 = 36
x² - 15x + 90-36 = 0
x² - 15x + 54 = 0
Solve by factoring:
(x-9) (x-6) = 0
x - 9=0 x - 6 = 0
x = 9 x = 6
The two polygons have sides of 6 and 9:
Hexagon = 6 sides
Nonagon = 9 sides
To check:
The sum of sides of two polygons is 15
6 + 9 = 15
The diagonals:
Polygon with 6 sides = 6 (6-3)
2
= 6(3)
2
= 18/2
= 9 diagonals
Polygon with 9 sides = 9 (9-3)
2
= 9 (6)
2
= 54/2
= 27 diagonals
The sum of the number of diagonals:
9 + 27 = 36
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