Solve sub-questions:

In the following figure, seg AB is the diameter of the circle with center P. Line CB be the tangent and line AC intersects a circle in point D. Prove that:

AC x AD = 4 (radius)²

Given: A circle with center P. CB tangent and line AC intersect a circle in point D

Construction: Join BD.

To Prove: ∴ ∠ADB =90° [Angle inscribed in semicircle]

∴ ∠PBC = 90° [Tangent perpendicular to the radius]

i.e. ∠ABC =90°

In Δ ACB and Δ ABD

∠ ABC = ∠ ADB [Each is of 90°]

∴ ∠ CAB = ∠DAB [Common angle]

∴ ΔACB ∼ ΔABD [AA property]

∴

∴AC × AD = (AB)²…(1)

AP = PB …(radii of the same circle)

∴ AB = AP +PB

∴ AB = 2AP

Substituting value of AB in equation (1)

AC × AD = (2AP)²

∴ AC × AD = 4(AP)²

∴ AC × AD =4 (radius)²