Class 10 Maths · Chapter 10

Circles

100 Class 10 Maths questions on Circles from the NCERT textbook, NCERT Exemplar and CBSE board papers 2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 5 of 5

1. Q81 CBSE 2026
   In the given ​figure, $AP$, $AQ$ and $BC$ ​are ​tangents ​to ​the circle with ​centre $O$. If $AB=6$ cm, $AC=7$ cm ​and $BC=5$ cm, then what is the length of $AP$?

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   $AP=9$ cm.

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2. Q82 CBSE 2026
   In the given figure, $TP$ is ​tangent ​to ​a circle with centre $O$. ​Diameter $BA$ when ​produced ​meets the ​tangent at $T$. If $\angle ABP=35^{\circ }$, then find the ​measure of $\angle PTA$.

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   $\angle PTA=20^{\circ }$.

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3. Q83 CBSE 2026
   Two concentric ​circles are of radii $5$ cm and $4$ cm. Find ​the ​length of ​the chord of the larger ​circle ​which touches the smaller circle.
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   The length of the required chord is $6$ cm.

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4. Q84 CBSE 2026
   In ​the ​given figure, $O$ is ​the centre of the ​circle. $PQ$ and $PR$ ​are ​tangents. ​Show that the quadrilateral $PQOR$ ​is cyclic.

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   Since $\angle OQP=\angle ORP=90^{\circ }$, the opposite angles at $Q$ and $R$ of quadrilateral $PQOR$ add up to $180^{\circ }$; hence $PQOR$ is a cyclic quadrilateral.

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5. Q85 CBSE 2026
   Prove ​that ​a ​parallelogram circumscribing ​a circle, is a ​rhombus.
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   Adding the equal tangent lengths at the four vertices gives $AB+CD=AD+BC$. Using $AB=CD$ and $AD=BC$ (parallelogram), this becomes $2AB=2AD$, so $AB=AD$. Hence $AB=BC=CD=DA$ and the parallelogram is a rhombus.

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6. Q86 CBSE 2026
   In the ​given ​figure, $△ABC$ is ​a right ​triangle in ​which $\angle B=90^{\circ }$, $AB=4$ cm and $BC=3$ cm. Find the radius ​of the circle inscribed in ​the triangle $ABC$.

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   The radius of the inscribed circle is $1$ cm.

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7. Q87 CBSE 2026
   A ​circle touches the side $BC$ of a $△ABC$ at $P$ ​and ​also ​touches the ​sides $AB$ and $AC$ ​produced at $Q$ and $R$ respectively. Prove that $AR=\frac{1}{2}(\text{perimeter of }△ABC)$.
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   $AR=\frac{1}{2}(\text{perimeter of }△ABC)$. By equal tangents $AQ=AR$, $BP=BQ$, $CP=CR$, so the perimeter $=AQ+AR=2AR$, giving $AR=\frac{1}{2}(\text{perimeter})$.

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8. Q88 CBSE 2026
   In the ​given figure, if a ​circle touches the side $QR$ of $△PQR$ at $S$ ​and extended sides $PQ$ and $PR$ at $M$ and $N$ ​respectively, then ​prove ​that ​: $PM=\frac{1}{2}(PQ+QR+PR)$

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   It is proved that $PM=\frac{1}{2}(PQ+QR+PR)$, using the equality of tangents drawn from the external points $P$, $Q$ and $R$.

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9. Q89 CBSE 2026
   Two tangents TP and TQ are drawn to ​a ​circle with ​centre O ​from an external ​point ​T. Prove that $\angle PTQ=2\angle OPQ$.
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   With $\angle PTQ=\theta$, the isosceles triangle $TPQ$ gives $\angle TPQ=90^{\circ }-\frac{\theta }{2}$, and since $\angle OPT=90^{\circ }$ we get $\angle OPQ=\frac{\theta }{2}$; therefore $\angle PTQ=2\angle OPQ$.

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10. Q90 CBSE 2026
    $AB$ ​is a chord ​of length $24$ ​cm of ​a circle of radius $15$ ​cm. The tangents at $A$ and $B$ intersect at a point $P$. Find ​the ​length $PA$.
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    $PA=20\text{cm}$.

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11. Q91 CBSE 2026
    Prove that ​the lengths of ​tangents drawn from ​an external ​point ​to a circle ​are equal.
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    In right triangles $OQP$ and $ORP$: $OQ=OR$ (radii), $OP$ is common and $\angle OQP=\angle ORP=90^{\circ }$, so $△OQP\cong △ORP$ by RHS; hence $PQ=PR$ by CPCT, i.e. the tangents drawn from an external point to a circle are equal in length.

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12. Q92 CBSE 2026
    Prove that ​a rectangle circumscribing a ​circle ​is a ​square.
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    $AB=BC=CD=DA$ and every angle is $90^{\circ }$, so the rectangle $ABCD$ is a square. Hence proved.

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13. Q93 CBSE 2026
    Prove ​that the tangent ​at ​any point ​of a ​circle is perpendicular ​to the radius through the point ​of contact.
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    For any point Q on the tangent XY other than the point of contact P, Q lies outside the circle, so $OQ=OR+RQ>OR=OP$. Hence OP is the shortest distance from O to XY, and therefore $OP\perp XY$ - the tangent at any point of a circle is perpendicular to the radius through the point of contact.

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14. Q94 CBSE 2026
    In the ​adjoining ​figure, $AB$ is the diameter of the circle ​with ​centre $O$. Two tangents $p$ and $q$ are drawn to ​the circle at points $A$ and $B$ ​respectively. Prove that $p\parallel q$. Further, a line $CD$ touches the circle ​at $E$ ​and $\angle BCD=110^{\circ }$. Find the measure of $\angle ADC$.

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    $p\parallel q$, because each tangent is perpendicular to the diameter $AB$ ($\angle DAB=\angle CBA=90^{\circ }$, so the co-interior angles are supplementary); and $\angle ADC=180^{\circ }-110^{\circ }=70^{\circ }$.

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15. Q95 CBSE 2026
    Two tangents $PA$ ​and $PB$ are drawn ​from an external ​point $P$ ​to a circle with centre $O$ as shown in the given ​figure. If $\angle OAB=15^{\circ }$, find ​the measures of ​each angle of $△PAB$.

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    $\angle PAB=75^{\circ }$, $\angle PBA=75^{\circ }$ and $\angle APB=30^{\circ }$.

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16. Q96 CBSE 2026
    In ​the given ​figure, $PA$ is ​the tangent to the circle with ​centre $O$ ​such that $OA=10$ cm, $AB=8$ ​cm and $AB\perp OP$. Find the length of $PB$.

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    $PB=\frac{32}{3}$ cm $\approx 10.67$ cm.

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17. Q97 CBSE 2026
    In the ​given figure, $TP$ and $TQ$ are tangents ​to a circle with centre $M$, touching another circle with centre $N$ at $A$ and $B$ respectively. It ​is given ​that $MQ=13$ cm, $NB=8$ cm, $BQ=35$ ​cm and $TP=80$ cm.

    1. (i)Name ​the quadrilateral $MQBN$.
    2. (ii)Is $MN$ parallel to $PA$ ? Justify your ​answer.
    3. (iii)Find ​length $TB$.
    4. (iv)Find length $MN$.

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    (i) $MQBN$ is a trapezium (a right trapezium, since $MQ\parallel NB$). (ii) No; $MP\parallel NA$ as both are perpendicular to $PT$, but $MP=13\ne 8=NA$, so $MPAN$ is a trapezium and $MN$ is not parallel to $PA$. (iii) $TB=45$ cm. (iv) $MN=25\sqrt{2}$ cm $\approx 35.36$ cm.

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18. Q98 CBSE 2026
    In ​the given figure, ​quadrilateral ​PQRS circumscribes the circle with centre O. Prove that the opposite sides ​of the quadrilateral PQRS subtend ​supplementary angles ​at the ​centre ​O.

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    Joining O to the vertices and to the points of contact A, B, C, D of sides PQ, QR, RS, SP, the SSS congruences such as $△OPA\cong △OPD$ give four equal pairs among the eight central angles ($\angle 1,\angle 2,\angle 3,\angle 4$ twice each). Since the eight angles total $360^{\circ }$, $\angle 1+\angle 2+\angle 3+\angle 4=180^{\circ }$, and regrouping gives $\angle POQ+\angle ROS=180^{\circ }$ and $\angle QOR+\angle SOP=180^{\circ }$ - the opposite sides of PQRS subtend supplementary angles at the centre O.

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19. Q99 CBSE 2026
    $PQ$ and $PR$ are two ​tangents to a circle ​with centre $O$ and ​radius $5$ cm. $AB$ ​is another tangent to the circle ​at $C$ which lies on $OP$. If $OP=13$ ​cm, then find the length $AB$ ​and $PA$.

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    $AB=\frac{20}{3}$ cm $\approx 6.67$ cm and $PA=\frac{26}{3}$ cm $\approx 8.67$ cm.

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20. Q100 CBSE 2026
    In a circular ​museum hall of ​radius $14$ ​m, some statues are displayed. Statues ​are kept inside the ​inner concentric circle of radius $7$ m. One such statue lying ​in sector $OAB$ is fenced along line ​segments $OA$, $AP$, $PB$ and $BO$, where $P$ ​is a point on the outer circle. Based on the above information, answer the following questions:

    1. (i)Find $m\angle AOP$.[1]
    2. (ii)Prove that $△OAP\cong △OBP$.[1]
    3. (iii)(a) Find the length of fencing required to protect the statue. (Take $\sqrt{3}=1.73$) OR (b) Find the area of quadrilateral $OAPB$. (Take $\sqrt{3}=1.73$)[2]
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    (i) $\angle AOP=60^{\circ }$. (ii) $△OAP\cong △OBP$ by RHS ($\angle OAP=\angle OBP=90^{\circ }$, $OA=OB$, $OP$ common). (iii)(a) Length of fencing $=14+14\sqrt{3}=38.22$ m, OR (b) Area of $OAPB=49\sqrt{3}=84.77$ m².

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