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Two Circles Touch Each Other Internally at a Point P. a Chord Ab of the Bigger Circle Intersects the Other Circle in C and D. Prove that `∠`Cpa = `∠`Dpb. - ICSE Class 10 - Mathematics

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ConceptTangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers

Question

Two circles touch each other internally at a point P. A chord AB of the bigger circle intersects the other circle in C and D. Prove that `∠`CPA = `∠`DPB.

Solution

Draw a tangent TS at P to the circles given.
Since TPS is the tangent, PD is the chord.
∴ `∠`PAB = `∠`BPS ……..(i) (Angles in alternate segment)
Similarly,
`∠`PCD = `∠`DPS …… (ii)
Subtracting (i) from (ii)
`∠`PCD - `∠`PAB = `∠`DPS - `∠`BPS
But in `∠`PAC,
Ext. `∠`PCD = `∠`PAB + `∠`CPA
∴ `∠`PAB + `∠`CPA - `∠`PAB  = `∠`DPS - `∠`BPS
⇒ `∠`CPA  = `∠`DPB

 

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Solution Two Circles Touch Each Other Internally at a Point P. a Chord Ab of the Bigger Circle Intersects the Other Circle in C and D. Prove that `∠`Cpa = `∠`Dpb. Concept: Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers.
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