SOLUTION: In each of the following, determine the number of sides of
a regular polygon with the stated property. If such a regular
polygon does not exist, explain why.
Each exterior angle
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Question 178210: In each of the following, determine the number of sides of
a regular polygon with the stated property. If such a regular
polygon does not exist, explain why.
Each exterior angle measures 25°.
The total number of diagonals is 4860.
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
In each of the following, determine the number of sides of
a regular polygon with the stated property. If such a regular
polygon does not exist, explain why.
-----------
1. Each exterior angle measures 25°.
The sum of the exterior angles is 360
number of sides= 360/25 = 14.4
That doesn't make any sense.
------------------------------------
The total number of diagonals is 4860.
nC2 - n is the number of diagonals of a regular polygon with n sides
nC2 - n = [n(n-1)]/[1*2] - n = (n^2-n)/2 - n
--------
(n^2-n)/2 - n = 4860
n^2 - n -2n = 9720
n^2 - 3n - 9720 = 0
solve for n
==============
Cheers,
Stan H.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Each exterior angle measures 25°
The sum of the exterior angles of any polygon is 360°. Therefore if a regular polygon has exterior angles of x°, then
must be an integer.
Not an integer, therefore there is no regular polygon with exterior angles measuring 25°
========================
The number of diagonals of a polygon is given by
 \over 2})
You are given that there are 4860 diagonals, so
 \over 2} = 4860)
Multiply by 2, remove parentheses, and put the equation in standard quadratic form:
Applying the quadratic formula:
 \pm sqrt{(-3)^2 - 4(1)(-9720)}}{2(1)} = \frac {3 \pm sqrt {38889}}{2} = \frac {3 \pm 3 sqrt{4321}}{2})
But one of these root is a negative number and both of them are irrational. Therefore there is no regular polygon with 4860 diagonals.
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