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