Class 10 Maths · Chapter 10

Circles

86 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 4 of 5

1. Q61 CBSE 2026
   In the ​given ​figure, $PA$ and $PB$ are ​tangents to a ​circle centred ​at $O$. ​If $\angle OAB=15^{\circ }$, ​then $\angle APB$ equals ​:

   1. (a)$30^{\circ }$
   2. (b)$15^{\circ }$
   3. (c)$45^{\circ }$
   4. (d)$10^{\circ }$
   Show answer

   (a) $30^{\circ }$

   Step-by-step solution →
2. Q62 CBSE 2026
   In ​the given ​figure, PA and PB are tangents to ​a circle ​centred at O. If $\angle AOB=130^{\circ }$, ​then $\angle APB$ ​is ​equal to :

   1. (a)$130^{\circ }$
   2. (b)$50^{\circ }$
   3. (c)$120^{\circ }$
   4. (d)$90^{\circ }$
   Show answer

   (b) $50^{\circ }$

   Step-by-step solution →
3. Q63 CBSE 2026
   $PQ$ is tangent ​to ​a circle ​with ​centre $O$. ​If $\angle POR=65^{\circ }$, then $m\angle PTR$ ​is

   1. (a)$65^{\circ }$
   2. (b)$58.5^{\circ }$
   3. (c)$57.5^{\circ }$
   4. (d)$45^{\circ }$
   Show answer

   (c) $57.5^{\circ }$

   Step-by-step solution →
4. Q64 CBSE 2026
   $PQ$ is tangent ​to ​a circle at a ​point $P$ on the circle. The number ​of tangents ​which can be ​drawn to the circle parallel to $PQ$, ​is

   1. (a)$2$
   2. (b)$1$
   3. (c)many
   4. (d)zero
   Show answer

   (b) $1$

   Step-by-step solution →
5. Q65 CBSE 2026
   Assertion (A): Radius is the smallest distance of a tangent from the centre of the circle.

   Reason (R): Radius is perpendicular to the tangent.

   1. (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
   2. (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
   3. (c)Assertion (A) is true, but Reason (R) is false.
   4. (d)Assertion (A) is false, but Reason (R) is true.
   Show answer

   (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

   Step-by-step solution →
6. Q66 CBSE 2026
   A circle ​is ​inscribed in a right triangle $ABC$, right angled ​at $B$. ​If the lengths of ​the two sides containing ​the right angle are $8$ cm and $15$ cm, find the radius of the incircle.
   Show answer

   The radius of the incircle is $3$ cm.

   Step-by-step solution →
7. Q67 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 $AP$?

   Show answer

   $AP=9$ cm.

   Step-by-step solution →
8. Q68 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 $\angle PTA$.

   Show answer

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

   Step-by-step solution →
9. Q69 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.
   Show answer

   The length of the required chord is $6$ cm.

   Step-by-step solution →
10. Q70 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.

    Show answer

    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.

    Step-by-step solution →
11. Q71 CBSE 2026
    Prove ​that ​a ​parallelogram circumscribing ​a circle, is a ​rhombus.
    Show answer

    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.

    Step-by-step solution →
12. Q72 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 $ABC$.

    Show answer

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

    Step-by-step solution →
13. Q73 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 $AR=\frac{1}{2}(\text{perimeter of }△ABC)$.
    Show answer

    $AR=\frac{1}{2}(\text{perimeter of }△ABC)$. By equal tangents $AQ=AR$, $BP=BQ$, $CP=CR$, so the perimeter $=AQ+AR=2AR$, giving $AR=\frac{1}{2}(\text{perimeter})$.

    Step-by-step solution →
14. Q74 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 ​: $PM=\frac{1}{2}(PQ+QR+PR)$

    Show answer

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

    Step-by-step solution →
15. Q75 CBSE 2026
    Two tangents TP and TQ are drawn to ​a ​circle with ​centre O ​from an external ​point ​T. Prove that $\angle PTQ=2\angle OPQ$.
    Show answer

    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 $\angle PTQ=2\angle OPQ$.

    Step-by-step solution →
16. Q76 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 $PA$.
    Show answer

    $PA=20\text{cm}$.

    Step-by-step solution →
17. Q77 CBSE 2026
    Prove that ​the lengths of ​tangents drawn from ​an external ​point ​to a circle ​are equal.
    Show answer

    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.

    Step-by-step solution →
18. Q78 CBSE 2026
    Prove that ​a rectangle circumscribing a ​circle ​is a ​square.
    Show answer

    $AB=BC=CD=DA$ and every angle is $90^{\circ }$, so the rectangle $ABCD$ is a square. Hence proved.

    Step-by-step solution →
19. Q79 CBSE 2026
    Prove ​that the tangent ​at ​any point ​of a ​circle is perpendicular ​to the radius through the point ​of contact.
    Show answer

    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.

    Step-by-step solution →
20. Q80 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 $\angle ADC$.

    Show answer

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

    Step-by-step solution →