A circle centered on the AC side of triangle ABC passes through vertex C and touches line AB

A circle centered on the AC side of triangle ABC passes through vertex C and touches line AB at point B. Find the diameter of the circle if AB = 15, AC = 25

Since AB is tangent to a circle centered at point O, and point B belongs to the circle, then triangle ABO is rectangular, since AB is tangent to the circle at point B.

Consequence: the radius of OB is perpendicular to the straight line AB.

By condition: AB = 15 cm, OB = OС = r, AO = AС – OС = 25 – r.

Consider a right-angled triangle ABO, applying the Pythagorean theorem:

AO ^ 2 = AB ^ 2 + OB ^ 2, (25 – r) ^ 2 = 15 ^ 2 + r ^ 2,

625 – 50 * r + r ^ 2 = 225 + r ^ 2,

50 * r = 625 – 225 = 400, r = 400: 50 = 8 (cm).

Let’s calculate the diameter of the circle: d = 2 * r = 2 * 8 = 16 (cm).