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Mathematics · Circles · NCERT Examples

Example 2

A complete, independently verified solution for NCERT Example 2.

Mathematics · Chapter 10 · NCERT Examples

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NCERT Example

Example 2

two tangents · angle proof

Two tangents TP and TQ are drawn from an external point T to a circle with centre O. Prove that ∠PTQ = 2∠OPQ.

Tangents TP and TQ from external point T to a circle with centre O.
Figure 10-9 from the supplied NCERT chapter.

Solution

Board-exam working:

Let ∠PTQ = θ.

Tangents from the same external point are equal.

TP=TQTP=TQ

Therefore, triangle TPQ is isosceles.

∠TPQ=∠TQP=180∘−θ2=90∘−θ2\angle TPQ=\angle TQP=\frac{180^\circ-\theta}{2}=90^\circ-\frac\theta2

The radius OP is perpendicular to tangent TP.

∠OPT=90∘\angle OPT=90^\circ
∠OPQ=∠OPT−∠TPQ\angle OPQ=\angle OPT-\angle TPQ
=90∘−(90∘−θ2)=θ2=90^\circ-\left(90^\circ-\frac\theta2\right)=\frac\theta2
θ=2∠OPQ\theta=2\angle OPQ
∴ ∠PTQ=2∠OPQ\therefore\ \angle PTQ=2\angle OPQ

Final answer

Hence proved: ∠PTQ = 2∠OPQ.