1952 AMC 12 Problem 21

Question

The sides of a regular polygon of

n

sides,

n\gt4,

are extended to form a star. The number of degrees at each point of the star is:

\dfrac{360}{n}

\dfrac{(n-4)180}{n}

\dfrac{(n-2)180}{n}

180-\dfrac{90}{n}

\dfrac{180}{n}

Answer: B

Concepts:regular polygon angle sum

Difficulty rating: 1500

Small hint:

Relate a star point to the exterior angle at each of its two base vertices

Big hint:

The angle at a star point and two exterior angles of the regular polygon form a straight-angle triangle

Solution:

Each exterior angle of a regular

n

-gon is

\frac{360}{n}

degrees. The two sides forming a point of the star, together with the intervening polygon side, make a triangle whose two base angles are those exterior angles. Hence its point angle is

180-2\left(\frac{360}{n}\right) =\frac{(n-4)180}{n}.

Thus, the correct answer is B.