SOLUTION: The ratio of diagonals to the sum of the measurebof polygons angles is 1to 26. Find the number of sides of the polygon

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Question 1151952: The ratio of diagonals to the sum of the measurebof polygons angles is 1to 26. Find the number of sides of the polygon
Answer by MathLover1(20850) About Me  (Show Source):
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the number of diagonals of an n-sided polygon is: n%28n+-+3%29+%2F+2
the sum of the measure of polygons angles is: 180%28n-2%29
if the ratio of diagonals to the sum of the measure of polygons angles is 1+%3A+26, we have
%28n%28n+-+3%29+%2F+2%29%3A%28180%28n-2%29%29=1%3A26
n%28n+-+3%29+%2F+%282%2A180%28n-2%29%29=1%3A26
26n%28n+-+3%29+=360%28n-2%29
26n%5E2+-+78n=360n-720
26n%5E2+-+78n-360n%2B720=0
26n%5E2+-+438n%2B720=0......factor
2%2813n%5E2+-+219n%2B360%29=0
2%2813n%5E2+-+24n-195n%2B360%29=0
2%28%2813n%5E2+-195n%29-+%2824n+-360%29%29=0
2%2813n%28n+-15%29-24+%28n+-15%29%29=0
2+%28n+-+15%29+%2813n+-+24%29+=+0-> we need only whole number as solution , and it is
%28n+-+15%29=+0=>n=15
the number of sides of the polygon is 15