In a right-angled triangle ABC, the height CH divides the hypotenuse AB into segments AH = 8 and BH = 18

In a right-angled triangle ABC, the height CH divides the hypotenuse AB into segments AH = 8 and BH = 18. The circle drawn on the segment CH, as on the diameter, intersects the sides AC and BC at points P and K. Find the length of the segment PK.

The РСК triangle is rectangular, since the angle is C = 90 and is inscribed in a circle, then РK is its diameter.

Triangles ВСН and АВС are similar in acute angle, angle СAН = ВСН.

Then in similar triangles, CH / BH = AH / CH.

CH ^ 2 = BH * AH = 18 * 8 = 144.

CH = 12 cm.

Then РK = CH = 12 cm.

Answer: The length of the РK segment is 12 cm.