ABC is a triangle right-angled at C. A semicircle is drawn on AB as diameter. Let P be any point on AC produced such that AP = AB = 10 cm. Further, B and P are joined. If BC = 8 cm, then what is BP equal to?

In right triangle ABC with right angle at C: AB = 10, BC = 8, so AC = √(100 - 64) = 6. P is on AC extended with AP = 10, so CP = AP - AC = 10 - 6 = 4. In right triangle BCP (angle BCP = 90° since P is on line AC extended and angle BCA = 90°): BP = √(BC² + CP²) = √(64 + 16) = √80 = 4√5 cm.