SOLUTION: The sum of the sides of two polygons is 9 and the sum of its diagonals is7. Find the numberof sides of its polygon

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Question 1125767: The sum of the sides of two polygons is 9 and the sum of its diagonals is7. Find the numberof sides of its polygon
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52802) About Me  (Show Source):
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From the condition, we have these two equations


m + n = 9,                (counting the sides)

%28m%2A%28m-3%29%29%2F2 + %28n%2A%28n-3%29%29%2F2 = 7    (counting the diagonals).



Equivalently

m + n = 9,                

m%2A%28m-3%29 + n%2A%28n-3%29 = 14.



Equivalently

m + n = 9,                

m%5E2-3m + n%5E2-3n = 14.
    


Equivalently

m + n = 9,                

m%5E2 + n%5E2 = 14 + 3*(m+n) = 14 + 3*9 = 14 +27 = 41.



Thus you have these two equations

m + n = 9,         (1)

m^2 + n^2 = 41.    (2)



From (1), express m = 9-n  and substitute it into (2).
You will get a quadratic equation. Solve it by any way you want.



Answer.  4 sides and 5 sides.



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Obviously if an algebraic solution is not required, you would solve this quickly with logical analysis.

The number of diagonals in a polygon with 3 sides is %283%2A0%29%2F2 = 0
The number of diagonals in a polygon with 4 sides is %284%2A1%29%2F2 = 2
The number of diagonals in a polygon with 5 sides is %285%2A2%29%2F2 = 5
The number of diagonals in a polygon with 6 sides is %286%2A3%29%2F2 = 9

If the total number of diagonals in two polygons is 7, then the polygons must have 4 and 5 sides.