if in a regular polygon the number of sides is increased by 3, each interior angle increases by 6. How many sides does the polygon have?

Question

Answer 1

Using “Turtle Graphics” (LOGO computer programming language), the turtle turns left n times (for n sides) and ends up at the starting point and pointing in the original direction. That is a total of 360 degrees.

So, if

 A = original interior angle

 N = original number of sides

AN = 360

Also,

 (A-6)(N+3) = 360    [Exterior angle decreases by 6]

That gives:

 AN + 3A – 6N - 18 = 360           [use F-O-I-L]

 360 + 3A – 6N – 18 = 360       [substitute 360 for AN]

 360 + 3(360/N) – 6N – 18 = 360    [substitute 360/N for A]

 1080/N – 6N – 18 = 0

  6N² + 18N - 1080 = 0      [multiply by (-N)]

   N² + 3N - 180 = 0     [divide by 6]

   (N+15)(N-12) = 0      [factor or use quadratic formula]

Either N = -15 or N = 12

    N = -15 makes no sense [number of sides]

    N = 12 is the original number of sides; It is expanded to 15 sides.