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In the figure (ii) given below, PQ is a tangent to the circle with centre O and AB is a diameter of the circle. If QA is parallel to PO, prove that PB is tangent to the circle.

In the figure (ii) given below, PQ is a tangent to the circle with centre O and AB is a diameter of the circle. If QA is parallel to PO, prove that PB is tangent to the circle. Circles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Circles

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Answer

Join OQ as shown in the figure below:

In the figure (ii) given below, PQ is a tangent to the circle with centre O and AB is a diameter of the circle. If QA is parallel to PO, prove that PB is tangent to the circle. Circles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

In △OAQ,

OA = OQ (Radius of the same circle.)

∠OAQ = ∠OQA.

Given QA || PO

∴ ∠OAQ = ∠POB (∵ corresponding angles are equal.)

and ∠OQA = ∠QOP (∵ alternate angles are equal.)

But ∠OAQ = ∠OQA,

∴ ∠POB = ∠QOP

Now in △OPQ and △OBP

OP = OP (Common sides)

OQ = OB (Radius of the same circle.)

∠QOP = ∠POB

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

As corresponding parts of congruent triangles are congruent,

∴ ∠OQP = ∠OBP

But ∠OQP = 90°

∴ ∠OBP = 90°

∴ PB is the tangent of the circle.

Hence, proved that PB is the tangent of the circle.

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