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From a point T outside a circle of centre O, tangents TP and TQ are drawn to the circle. Prove that OT is the right bisector of line segment PQ. - CBSE Class 10 - Mathematics

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Question

From a point T outside a circle of centre O, tangents TP and TQ are drawn to the circle. Prove that OT is the right bisector of line segment PQ.

Solution

TP and TQ are tangents drawn from an external point T to the circle. O is the centre of the circle.

Suppose OT intersect PQ at point R.

In ∆OPT and ∆OQT,

OP = OQ      (Radii of the circle)

TP = TQ       (Lengths of tangents drawn from an external point to a circle are equal.)

OT = OT       (Common sides)

∴ ∆OPT ≅ ∆OQT  (By SSS congruence rule)

So, PTO = QTO    (By CPCT)      .....(1)

Now, in ∆PRT and ∆QRT,

TP = TQ                  (Lengths of tangents drawn from an external point to a circle are equal.)

PTO = QTO       [From (1)]

RT = RT                  (Common sides)

∴ ∆PRT ≅ ∆QRT    (By SAS congruence rule)

So, PR = QR      .....(2)        (By CPCT)

And, PRT = QRT    (By CPCT)

Now,

PRT + QRT = 180°         (Linear pair)

⇒ 2PRT = 180°

PRT = 90°

PRT = QRT = 90°   .....(3)

From (2) and (3), we can conclude that

OT is the right bisector of the line segment PQ.

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Solution From a point T outside a circle of centre O, tangents TP and TQ are drawn to the circle. Prove that OT is the right bisector of line segment PQ. Concept: Circles Examples and Solutions.
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