How many sides does a regular polygon have if each interior angle is three times the measure of each exterior angle measure?(2 points)
Let's denote the measure of each exterior angle as x. According to the question, each interior angle is three times the measure of the exterior angle. Therefore, the measure of each interior angle is 3x.
In a polygon with n sides, the sum of the interior angles is (n - 2) * 180 degrees. Since each interior angle is 3x, we can write the equation:
(n - 2) * 180 = 3x * n
Simplifying the equation, we get:
180n - 360 = 3xn
180n - 3xn = 360
Factorizing the equation, we have:
n(180 - 3x) = 360
n = 360 / (180 - 3x)
Since n represents the number of sides in a polygon, it must be a positive integer. Hence, for n to be an integer, the denominator (180 - 3x) must be a factor of 360.
Let's find the factors of 360:
Factors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
Now, let's substitute these factors into the equation n = 360 / (180 - 3x) and find the corresponding values of n:
For n = 360 / (180 - 3x), when x = 20, n ≈ 3.16 (not an integer)
For n = 360 / (180 - 3x), when x = 30, n ≈ 4 (an integer)
Therefore, a regular polygon with each interior angle three times the measure of each exterior angle has 4 sides.