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Mathematics · Circles

NCERT Examples

All 3 worked examples from Circles, with independently verified solutions and direct navigation.

Mathematics · Chapter 10 · NCERT Examples

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Verified NCERT questions and solutionsSolutions are checked independently and follow the supplied NCERT chapter edition.

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NCERT Example

Example 1

concentric circles · perpendicular chord theorem

Prove that in two concentric circles, a chord of the larger circle that touches the smaller circle is bisected at the point of contact.

Two concentric circles with chord AB of the larger circle tangent to the smaller circle at P.
Figure 10-8 from the supplied NCERT chapter.

Solution

Board-exam working:

Let the two circles have common centre O. Chord AB of the larger circle touches the smaller circle at P.

Join OP.

Since AB is tangent to the smaller circle at P, its radius OP is perpendicular to AB.

OP⊥ABOP\perp AB

AB is also a chord of the larger circle. The perpendicular from the centre of a circle to a chord bisects the chord.

AP=PBAP=PB

Final answer

Hence, the chord AB is bisected at P.

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.

NCERT Example

Example 3

chord · tangent length

PQ is a chord of length 8 cm in a circle of radius 5 cm. The tangents at P and Q meet at T. Find TP.

Chord PQ of length 8 centimetres in a circle of radius 5 centimetres, with tangents meeting at T.
Figure 10-10 from the supplied NCERT chapter.

Solution

Board-exam working:

Join OT and let it meet PQ at R.

Since TP = TQ, triangle TPQ is isosceles. OT bisects ∠PTQ and is perpendicular to PQ.

PR=RQ=82=4 cmPR=RQ=\frac82=4\text{ cm}

In right triangle OPR:

OR=OP2−PR2OR=\sqrt{OP^2-PR^2}
=52−42=3 cm=\sqrt{5^2-4^2}=3\text{ cm}

Triangles TRP and PRO are similar by AA similarity.

TPPO=PRRO\frac{TP}{PO}=\frac{PR}{RO}
TP5=43\frac{TP}{5}=\frac43
TP=203 cmTP=\frac{20}{3}\text{ cm}

Final answer

TP=203 cmTP=\frac{20}{3}\text{ cm}