Given AB = 45 AO = 75 AB-tangent, AO secant. Find the radius.

The presence of the word “radius” in the statement of the task indicates that here, for sure, we are talking about a circle or a circle with a center at point O. This is also evidenced by the conditions: tangent AB = 45, which is less than 75 = AO – secant.
As you know, the tangent to the circle is perpendicular to the radius drawn to the tangent point. This means that triangle ABO is a right-angled triangle, where are known: the length of the hypotenuse AO = 75 and the length of one of the legs – AB = 45. By the Pythagorean theorem, we find the length of the unknown leg OB, which is the radius of the circle. We have: AO² = AB² + ОВ², whence ОВ² = AO² – AB² = 75² – 45² = (75 – 45) * (75 + 45) = 30 * 120 = 3600. Therefore, ОВ = √ (3600) = 60.
Answer: 60.