Problem No. 9. Points on a circle

How many points can be drawn on a circle such that each two adjacent points are at the same distance, and each three adjacent points form an angle of $7^o 30'$.

Student tip

You need to recall on the properties of inscribed angles in a circle.

Solution

Let A, B and C be the three consecutive points on the circle. Then, we have that the inscribed angles ACB and BAC have $7^o 30'$ each, since the distance between A and B is equal to the distance between B and C. According to the relationship between central angle and inscribed angle, we have that the central angle that corresponds to the inscribed angle ACB is $2 \cdot 7^o 30' = 15^o$. The same stands for the central angle that corresponds to the arc between any two adjusted points. Since, $360^o / 15^o = 24$, we have that there are 24 points on a circle with the given property.