Find the Length of the Tangent of the Semicircle
Geometry math problem
Find the length of the tangent of a small semicircle which is inscribed inside another semicircle

From the figure, BC and PQ are tangents of the smaller semicircle which is inscribed inside another semicircle. PQ = 5 cm and PB = 4 cm, then find the length of the tangent PQ
Solution
Connect C with the centre of the smaller semicircle
Then we get a right triangle OBC

Let the radius of the smallest semicircle = r and PQ = k
Apply Pythagoras theorem in the triangle OBC
OB2 = OC2 + BC2
⇒ ( 4+r )2 = ( r )2 + 52
⇒ 16 + 8r + r2 = r2 + 25
⇒ 8r = 9
⇒ r = 9/8 cm
Apply intersecting chords theorem, then
AP × PB = PQ2
⇒ 2r × 4 = k2
⇒ 2( 9/8 ) × 4 = k2
⇒ k2 = 9
⇒ k = 3 cm
⇒ Length of the tangent, PQ = 3 cm