JOIN NOW

Mathematics

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

11 Likes

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.

Answered By

9 Likes

Related Questions