(20點) F.2-3 Maths Question

Sum of interior angles of a polygon = 180 (n - 2), where n is the number of sides

For a n-sided regular polygon, there are n interior angles and each angle is the same.
each angle = [180 (n - 2)] / n. ------------(1)

According to the question:

Let N be the number of sides of the original regular polygon,
Using (1), each angle = [180 (N - 2)] /N = (180N - 360) / N ----------------(2)

If the number of sides of the polygon is doubled, there are 2N sides.
Using (1), each new angle
= [180 (2N - 2)] /2N
= [90 (2N - 2)] / N
= (180N - 180) / N ----------------------(3)

As each interior angle would increase by 10˚, from (2) and (3)
(180N - 180) / N - (180N - 360) / N = 10
(180N - 180 - 180N + 360) / N = 10 (same denominator)
180 / N = 10
180 = 10N
N = 18

Hence, the original polygon has 18 sides.