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From a point outside a circle, with centre O, tangents PA and PB are drawn. Prove that

(i) ∠AOP = ∠BOP

(ii) OP is the perpendicular bisector of the chord AB.

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Answer

The figure is shown below:

From a point outside a circle, with centre O, tangents PA and PB are drawn. Prove that (i) ∠AOP = ∠BOP (ii) OP is the perpendicular bisector of the chord AB. Circles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

(i) In △AOP and △BOP,

OP = OP (Common sides)

OA = OB (Radius of the circle)

∠OAP = ∠OBP (Both are equal to 90° as tangents and radius on point of contact are perpendicular to each other.)

∴ △OAP ≅ △OBP (S.A.S. axiom of congruency)

(As Corresponding parts of congruent triangles are congruent)

∴ ∠AOP = ∠BOP and ∠APO = ∠BPO.

Hence, proved that ∠AOP = ∠BOP.

(ii) In △APM and △BPM,

PM = PM (Common side)

∠APM = ∠BPM (Proved above)

AP = BP (∵ tangents from an exterior point to a circle are equal in length)

∴ △APM ≅ △BPM (S.A.S. axiom of congruency)

(Congruent parts of congruent triangles are congruent.)

∴ AM = BM and ∠AMP = ∠BMP

But ∠AMP + ∠BMP = 180°

∴ ∠AMP = ∠BMP = 90°.

Hence, proved that OP is perpendicular bisector of AB at M.

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