Q. 35.0( 2 Votes )

# AP and AQ are two tangents drawn from an external point A to a circle with centre O, P and Q are points of contact. If PR is a diameter, let us prove that OA ∥ RQ

Answer :

In Δ APO and Δ AQO

AP = AQ (Tangents drawn to the same circle from an external point are always equal in length)

AO = AO (Common)

∠ OPA = ∠ OQA (Tangents are perpendicular to the line joining the centre)

So Δ APO and Δ AQO are congruent by S.A.S. axiom of congruency.

So from corresponding parts of congruent triangle we get

∠ POA = ∠ QOA = x (Let)

∠ POA + ∠ QOA = 2x

∠ QOR = 180^{0}-2x (Angle on a straight line)

∠ OQR = ∠ ORQ (Δ OQR is isosceles)

∠ OQR + ∠ ORQ = 180^{0}-(180^{0}-2x) = 2x

Since they are equal so

∠ OQR = x = ∠ AOQ

∠ OQR and ∠ AOQ forms a pair of alternate interior angles

So OA∥ RQ

Rate this question :

Let us prove that for a quadrilateral circumscribed about a circle, the angles subtended by any two opposite sides at the centre are supplementary to each other.

West Bengal Board - MathematicsThree equal circles touch one another externally. Let us prove that the centres of the three circles form an equilateral triangle.

West Bengal Board - MathematicsIn the adjoining figure O is the centre and BOA is a diameter of the circle. A tangent drawn to a circle at the point P intersects the extended BA at the point T. If ∠PBO = 30°, let us find the value of ∠PTA.

West Bengal Board - Mathematics

The tangent drawn at points P and Q on the circumference of a circle intersect at A. If ∠PAQ = 60°, let us find the value of ∠APQ.

West Bengal Board - MathematicsAP and AQ are two tangents drawn from an external point A to a circle with centre O, P and Q are points of contact. If PR is a diameter, let us prove that OA ∥ RQ

West Bengal Board - MathematicsIn the adjoining figure, ΔABC circumscribed a circle and touches the points P,Q,R. If AP = 4 cm, BP = 6cm, AC = 12 cm and BC = x cm, let us determine the value of x.

West Bengal Board - Mathematics

Two circles touch each other externally at the point C. A direct common tangent AB touch the two circle at the points A and B. Value of ∠ACB is

West Bengal Board - Mathematics