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Two Circle with Centres O and O ' Are Drawn to Intersect Each Other at Points a and B.Prove that Oa Bisects Angle Bac. - 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 circle with centres O and O ' are drawn to intersect each other at points A and B.
Centre O of one circle lies on the circumference of the other circle and CD is drawn tangent to the circle with centre O ' at A. prove that OA bisects angle BAC.

Solution

Join OA, OB, O'A, O'B and O'O.
CD is the tangent and AO is the chord.
`∠`OAC = `∠`OBA (angles in alternate segment)
In ΔOAB ,
OA = OB (Radii of the same circle)
∴ OAB = `∠`OBA …..(ii)
From (i) and (ii)
`∠`OAC = `∠`OAB
Therefore, OA is bisector of `∠`BAC

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Solution Two Circle with Centres O and O ' Are Drawn to Intersect Each Other at Points a and B.Prove that Oa Bisects Angle Bac. Concept: Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers.
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