From point A to the circle with center O, a tangent line AK is drawn and intersecting AO, intersecting

From point A to the circle with center O, a tangent line AK is drawn and intersecting AO, intersecting the circle at M and N. Find the angle KMN if the angle KAN = 38 degrees.

The radius OK, drawn to the point of tangency K, is perpendicular to the tangent AK, then the triangle AOK is rectangular, in which the angle AOK = MOK = (90 – 38) = 52.

In the MOK triangle, MO = OK = R, then the MOK triangle is isosceles with the base KM, and therefore the angle OMC = OKM = (180 – MOK) / 2 = (180 – 52) / 2 = 64.

KMН angle = OMK = 64.

Answer: Angle KMН = 64.