# The circle centered on side AC of triangle ABC passes through vertex C and touches line AB at point B.

The circle centered on side AC 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.

Let’s draw the segment OB to the point of tangency of the circle. Since AB is tangent, the angle OBA is rectangular.
In the VOS triangle, the OB and OS segments are equal to the radius of the circle. Since, by condition, the length of the segment AC = 25 cm, then the length of the segment OA = AC – OC = 25 – R.
In a right-angled triangle AOB, by the Pythagorean theorem, we define the leg OB.
ОВ2 = ОА2 – АВ2.
R2 = (25 – R) 2 – 152.
R2 = 625 – 50 * R + R2 – 225.
50 * R = 400.
R = 400/50 = 8 cm.
Then D = 2 * R = 16 cm.

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