From a point P outside the circle, with centre O, tangents PA and PB are drawn. Prove that: (i) ∠AOP = ∠BOP (ii) OP is the ⊥ bisector of chord AB.

The circle with centre O and tangents PA and PB drawn from point P outside the circle is shown in the below figure: (i) In ∆AOP and ∆BOP, we have ⇒ AP = BP [Tangents from an exterior point P, are equal in length.] ⇒ OA = OB [Radius of circle] ⇒ OP = OP [Common] ∴ ΔAOP ≅ ΔBOP by SSS axiom. ∴ ∠AOP = ∠BOP [By C.P.C.T.] Hence, proved that ∠AOP = ∠BOP. (ii) In ∆OAM and ∆OBM, we have OA = OB [Radii of the same circle] ∠AOM = ∠BOM [Proved ∠AOP = ∠BOP] OM = OM [Common] ∴ ΔOAM ≅ ΔOBM [By SAS axiom] By C.P.C.T. ⇒ AM = MB and ∠OMA = ∠OMB = x From figure, ⇒ ∠OMA + ∠OMB = 180° [As, AB is a straight line] ⇒ x + x = 180° ⇒ 2x = 180° ⇒ x = ⇒ x = 90°. Thus, ∠OMA = ∠OMB = 90°. Hence proved, that OP is the ⊥ bisector of chord AB.

In the given figure, if AB = AC then prove that BQ = CQ.

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Tangents AP and AQ are drawn to a circle, with center O, from an exterior point A. Prove that :
∠PAQ = 2∠OPQ

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In the given figure, two circles touch each other externally at point P. AB is the direct common tangent of these circles. Prove that :
(i) tangent at point P bisects AB.
(ii) angle APB = 90°.

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Radii of two circles are 6.3 cm and 3.6 cm. State the distance between their centers if :
(i) they touch each other externally,
(ii) they touch each other internally.

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Mathematics

From a point P outside the circle, with centre O, tangents PA and PB are drawn. Prove that:
(i) ∠AOP = ∠BOP
(ii) OP is the ⊥ bisector of chord AB.

Circles

Related Questions

In the given figure, if AB = AC then prove that BQ = CQ.

Tangents AP and AQ are drawn to a circle, with center O, from an exterior point A. Prove that :
∠PAQ = 2∠OPQ

In the given figure, two circles touch each other externally at point P. AB is the direct common tangent of these circles. Prove that :
(i) tangent at point P bisects AB.
(ii) angle APB = 90°.

Radii of two circles are 6.3 cm and 3.6 cm. State the distance between their centers if :
(i) they touch each other externally,
(ii) they touch each other internally.

Mathematics

From a point P outside the circle, with centre O, tangents PA and PB are drawn. Prove that:(i) ∠AOP = ∠BOP(ii) OP is the ⊥ bisector of chord AB.

Circles

Related Questions

In the given figure, if AB = AC then prove that BQ = CQ.

Tangents AP and AQ are drawn to a circle, with center O, from an exterior point A. Prove that :∠PAQ = 2∠OPQ

In the given figure, two circles touch each other externally at point P. AB is the direct common tangent of these circles. Prove that :(i) tangent at point P bisects AB.(ii) angle APB = 90°.

Radii of two circles are 6.3 cm and 3.6 cm. State the distance between their centers if :(i) they touch each other externally,(ii) they touch each other internally.

From a point P outside the circle, with centre O, tangents PA and PB are drawn. Prove that:
(i) ∠AOP = ∠BOP
(ii) OP is the ⊥ bisector of chord AB.

Tangents AP and AQ are drawn to a circle, with center O, from an exterior point A. Prove that :
∠PAQ = 2∠OPQ

In the given figure, two circles touch each other externally at point P. AB is the direct common tangent of these circles. Prove that :
(i) tangent at point P bisects AB.
(ii) angle APB = 90°.

Radii of two circles are 6.3 cm and 3.6 cm. State the distance between their centers if :
(i) they touch each other externally,
(ii) they touch each other internally.