Question

In the figure; PA is a tangent to the circle, PBC is secant and AD bisects angle BAC. Show that triangle PAD is an isosceles triangle. Also, show that:

`∠CAD = 1/2 (∠PBA - ∠PAB)`

Answer 1

i. PA is the tangent and AB is a chord

∴ ∠PAB = ∠C  ...(i) (Angles in the alternate segment)

AD is the bisector of ∠BAC

∴ ∠1 = ∠2  ...(ii)

In ΔADC,

Ext. ∠ADP = ∠C + ∠1

`=>` Ext ∠ADP = ∠PAB + ∠2 = ∠PAD

Therefore, ΔPAD is an isosceles triangle.

ii. In ΔABC,

Ext. ∠PBA = ∠C + ∠BAC

∴ ∠BAC = ∠PBA – ∠C

`=>` ∠1 + ∠2 = ∠PBA – ∠PAB  ...(From (i) part)

`=>` 2∠1 = ∠PBA – ∠PAB

`=> ∠1 = 1/2 (∠PBA - ∠PAB)`

`=> ∠CAD = 1/2 (∠PBA - ∠PAB)`