ICSE Class 10CISCE
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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)` - ICSE Class 10 - Mathematics

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ConceptTangent 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

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)`

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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.
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