#### 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)`

#### Solution

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 Δ ADC,

Ext. `∠`ADP = `∠`C + `∠`1

⇒ Ext `∠`ADP = `∠`PAB + `∠`2 = `∠`PAD

Therefore, Δ PAD is an isosceles triangle.

ii) In ΔABC,

Ext. `∠`PBA = `∠`C + `∠`BAC

`∠`BAC = `∠`PBA - `∠`C

⇒ `∠``1 + `∠`2 = `∠`PBA - `∠`PAB

(fom (i) part)

`2∠1 = ∠PBA - ∠PAB`

`∠1=1/2(∠PBA - ∠PAB)`

⇒`∠CAD = 1/2 (∠PBA - ∠ PAB)`

Is there an error in this question or solution?

Solution 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)` Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.