If the difference between the exterior regular polygon and an (n+1) sided regular polygon is 12 find the value of n.
To find the value of n, we first need to understand the relationship between the exterior regular polygon and the (n+1)-sided regular polygon.
The exterior angle of a regular polygon is the angle formed by extending one side of the polygon. In a regular n-sided polygon, the exterior angle is 360°/n.
Now, let's denote the number of sides of the exterior regular polygon as m. The exterior angle of this polygon is 360°/m.
The difference between the two polygons is given as 12. This means that the exterior angle of the (n+1)-sided regular polygon is 12 degrees less than the exterior angle of the m-sided regular polygon.
Therefore, we can set up the following equation:
360°/m - 360°/(n+1) = 12
To find the value of n, we can solve this equation.
Let's simplify the equation:
(360(n+1) - 360m)/(m(n+1)) = 12
Expanding and rearranging:
360n + 360 - 360m = 12m(n+1)
Collecting like terms:
360n - 360m + 360 - 12m(n+1) = 0
Simplifying further:
360(n - m + 1) - 12m(n+1) = 0
Factoring out the common term:
(360 - 12m)(n - m + 1) = 0
To satisfy this equation, either (360 - 12m) = 0 or (n - m + 1) = 0.
1. If (360 - 12m) = 0, we can solve for m:
360 - 12m = 0
12m = 360
m = 360/12
m = 30
2. If (n - m + 1) = 0, we can solve for n:
n - 30 + 1 = 0
n - 29 = 0
n = 29
Therefore, the value of n is 29.
The wording looks jumbled
The exterior angle of an n-gon is 360/n
and the exterior angle of an (n+1)-gon is 360/(n+1)
so 360/n - 360(n+1) = 12
times n(n+1) , the LCD
360(n+1) - 360n = 12n(n+1)
360n + 360 - 360n = 12n^2 + 12n
12n^2 + 12n - 360 = 0
n^2 + n - 30 = 0
(n+6)(n-5) = 0
n = 5 or n = a negative
n = 5
check:
the exterior angle of a regular pentagon is 72°
the exterior angle of a hexagon is 60°
which is a difference of 12°