Question 97
The number of diagonals to a regular polygon A is given as 20. What is the exterior angle of the regular polygon A?
Solution
The number of diagonals formula is given as $$\dfrac{n\left(n\ -\ 3\right)}{2}$$. The number of diagonals is given as 20 in the questions. The number of sides can be calculated as.
$$\dfrac{n\left(n\ -\ 3\right)}{2}\ =\ 20$$
$$n\left(n\ -\ 3\right)\ =\ 40$$
$$n^2\ -3n\ -\ 40\ =\ 0$$
$$n^2\ -\ 8n+\ 5n\ -\ 40\ =\ 0$$
$$\left(n\ +\ 5\right)\left(n\ -\ 8\right)\ =\ 0$$
$$n\ =\ 8\ or\ n\ =\ -5$$
Since n cannot be negative, the value of n is 8.
The exterior angle can be calculated as $$\dfrac{360}{n}\ =\ \dfrac{360}{8}\ =\ 45^{\circ\ }$$
The correct answer is option B.
