A circle is described around a regular triangle ABC, The length of the arc AB is equal to 2 p (pi)

A circle is described around a regular triangle ABC, The length of the arc AB is equal to 2 p (pi) Find: a) the radius of this circle b) the length of one of the medians of the triangle.

From the point O, the center of the circle, we construct the radii OA and OB.

Degree measure of the inscribed angle AOB = 120.

From the formula for the length of an arc of a circle, we express its radius.

L = π * R * AOB / 180.

R = L * 180 / π * AOB = 2 * π * 180 / π * 120 = 3 cm.

ОА = ОВ = 3 cm.

Point O is the point of intersection of the medians of the regular triangle ABC, which divides the median BD in a ratio of 2/1.

ОВ / ОD = 2/1.

OD = ОВ / 2 = 3/2 = 1.5 cm.

Then the median ВD = ОВ + ОD = 3 + 1.5 = 4.5 cm.

Answer: The radius of the circle is 3 cm, the median of the triangle is 4.5 cm.