Respuesta :
The interior angle of a polygon, and its exterior angle form a linear pair.
- The value of n is 8.
Reasons:
The given parameters of the polygon are;
The difference in the exterior angles of the polygon = 5°
Number of sides on one of the polygon = n
Number of sides on the other polygon = n + 1
We get;
The interior angles of a regular polygon are given by the following formula;
[tex]\theta_1 = \mathbf{\dfrac{(n - 2) \times 180^{\circ}}{n}}[/tex]
Which gives;
[tex]\theta_2 = \mathbf{\dfrac{((n + 1) - 2) \times 180^{\circ}}{n + 1}}[/tex]
[tex]\mathbf{\left(180 - \dfrac{(n - 2) \times 180^{\circ}}{n} \right) - \left( 180 - \dfrac{((n + 1) - 2) \times 180^{\circ}}{n + 1} \right)} = 5[/tex]
Simplifying with a graphing calculator gives;
[tex]\dfrac{360}{n^2 + n} = 5[/tex]
5·(n² + n) = 360
5·n² + 5·n - 360 = 0
n² + n - 72 = 0
(n - 8)·(n + 9) = 0
Therefore;
The number of sides on on of the polygon, n = 8
- The value of n = 8
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