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 1 of 5

1. Q1 NCERT · Exercise 10.1
   How many tangents ​can ​a ​circle ​have?
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   A circle has infinitely many tangents, one at each of its infinitely many points.

   Step-by-step solution →
2. Q2 NCERT · Exercise 10.1
   Fill in ​the blanks:

   1. (i)A tangent to a circle intersects it in ____ point(s).
   2. (ii)A line intersecting ​a circle in two points is ​called ​a ____. ​
   3. (iii)A circle ​can have ​____ parallel tangents at the ​most.
   4. (iv)The common point of a tangent to a circle and the circle is called ____.
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   (i) One (ii) secant (iii) two (iv) point of contact.

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3. Q3 NCERT · Exercise 10.1
   A tangent $PQ$ at ​a point $P$ ​of a circle of ​radius ​5 cm ​meets ​a line through the centre $O$ at a ​point $Q$ ​so that $OQ=12$ cm. Length $PQ$ is :

   1. (a)12 cm
   2. (b)13 cm
   3. (c)8.5 cm
   4. (d)$\sqrt{119}$ cm
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   (d) $\sqrt{119}$ cm

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4. Q4 NCERT · Exercise 10.1
   Draw ​a circle and two ​lines ​parallel to a given line such that ​one is ​a tangent and the other, ​a secant to ​the circle.
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   Drop the perpendicular from $O$ to $l$ meeting the circle at $P$. The line through $P$ parallel to $l$ is the required tangent ($d=r$), and a line parallel to $l$ through any point $N$ on $OP$ with $ON<r$ is the required secant ($d<r$).

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5. Q5 NCERT · Exercise 10.2
   Prove ​that the tangents ​drawn at the ends ​of a ​diameter of ​a ​circle are parallel.
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   Since $\angle PAB=\angle ABS=90^{\circ }$ and these are alternate angles made by the transversal $AB$, the tangents $PQ$ and $RS$ at the ends of the diameter $AB$ are parallel. Hence proved.

   Step-by-step solution →
6. Q6 NCERT · Exercise 10.2
   Prove that the angle between the two tangents drawn from an external point ​to a circle is supplementary to the angle subtended by ​the line-segment joining the points ​of contact at ​the centre.
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   $\angle APB+\angle AOB=180^{\circ }$, so the angle between the tangents is supplementary to the angle the chord of contact subtends at the centre. Hence proved.

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7. Q7 NCERT · Exercise 10.2
   Prove ​that ​opposite ​sides of ​a ​quadrilateral circumscribing a circle subtend ​supplementary angles at ​the centre of the circle.
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   $\angle AOB+\angle COD=180^{\circ }$ and $\angle BOC+\angle AOD=180^{\circ }$, so opposite sides of the circumscribing quadrilateral subtend supplementary angles at the centre. Hence proved.

   Step-by-step solution →
8. Q8 NCERT · Exercise 10.2
   From a ​point Q, the ​length of the tangent to a circle is 24 ​cm and the distance ​of Q ​from the centre is 25 ​cm. The radius of ​the circle is

   1. (a)7 cm
   2. (b)12 cm
   3. (c)15 cm
   4. (d)24.5 cm
   Show answer

   (a) 7 cm

   Step-by-step solution →
9. Q9 NCERT · Exercise 10.2
   In Fig. 10.11, ​if $TP$ and $TQ$ ​are the two ​tangents to a circle with centre $O$ so ​that $\angle POQ=110^{\circ }$, ​then $\angle PTQ$ is ​equal ​to

   1. (a)$60^{\circ }$
   2. (b)$70^{\circ }$
   3. (c)$80^{\circ }$
   4. (d)$90^{\circ }$
   Show answer

   (b) $70^{\circ }$

   Step-by-step solution →
10. Q10 NCERT · Exercise 10.2
    If tangents ​PA and PB from a ​point P to a ​circle with ​centre O ​are inclined to each other ​at angle of $80^{\circ }$, then $\angle POA$ is equal ​to

    1. (a)$50^{\circ }$
    2. (b)$60^{\circ }$
    3. (c)$70^{\circ }$
    4. (d)$80^{\circ }$
    Show answer

    (a) $50^{\circ }$

    Step-by-step solution →
11. Q11 NCERT · Exercise 10.2
    Prove ​that the perpendicular at the point ​of ​contact to ​the ​tangent to a ​circle ​passes through ​the centre.
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    The perpendicular to the tangent at the point of contact $P$ coincides with the radius $OP$, and therefore passes through the centre $O$. Hence proved.

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12. Q12 NCERT · Exercise 10.2
    The ​length of ​a ​tangent from ​a point $A$ at distance $5$ cm from the ​centre ​of the circle is $4$ cm. Find the radius of the ​circle.
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    Radius $r=3$ cm.

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13. Q13 NCERT · Exercise 10.2
    Two concentric circles are ​of radii $5$ ​cm ​and $3$ cm. Find the ​length of ​the chord of the ​larger circle ​which touches the smaller ​circle.
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    Length of the chord $=8$ cm.

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14. Q14 NCERT · Exercise 10.2
    A ​quadrilateral $ABCD$ ​is drawn to circumscribe a ​circle (see ​Fig. ​10.12). Prove ​that $AB+CD=AD+BC$.

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    $AB+CD=AD+BC$. Hence proved.

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15. Q15 NCERT · Exercise 10.2
    In ​Fig. ​10.13, ​XY and ​X'Y' are two ​parallel tangents to ​a circle with centre O and another tangent AB ​with point of contact C intersecting XY ​at A and X'Y' at B. Prove that $\angle AOB=90^{\circ }$.

    Show answer

    $\angle AOB=90^{\circ }$. Hence proved.

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16. Q16 NCERT · Exercise 10.2
    Prove ​that ​the ​parallelogram circumscribing ​a circle is a ​rhombus.
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    All four sides are equal, $AB=BC=CD=DA$, so the circumscribing parallelogram $ABCD$ is a rhombus. Hence proved.

    Step-by-step solution →
17. Q17 NCERT · Exercise 10.2
    A triangle ABC ​is drawn to circumscribe ​a ​circle ​of radius 4 cm such that the segments BD and DC into which ​BC is divided by the ​point ​of ​contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.

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    $AB=15$ cm and $AC=13$ cm.

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18. Q18 Exemplar · Exercise 9.1
    If ​radii of ​two ​concentric circles are ​4 ​cm ​and 5 ​cm, then the length of each chord of ​one circle which is tangent to the other circle is

    1. (a)$3$ cm
    2. (b)$6$ cm
    3. (c)$9$ cm
    4. (d)$1$ cm
    Show answer

    (b) $6$ cm

    Step-by-step solution →
19. Q19 Exemplar · Exercise 9.1
    In ​Fig. 9.3, ​if $\angle AOB=125^{\circ }$, then $\angle COD$ ​is ​equal to

    1. (a)$62.5^{\circ }$
    2. (b)$45^{\circ }$
    3. (c)$35^{\circ }$
    4. (d)$55^{\circ }$
    Show answer

    (d) $55^{\circ }$

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20. Q20 Exemplar · Exercise 9.1
    In Fig. 9.4, $AB$ is ​a ​chord of the circle ​and $AOC$ is its diameter such that $\angle ACB=50^{\circ }$. If $AT$ is the tangent to the ​circle ​at ​the point $A$, ​then $\angle BAT$ ​is equal to

    1. (a)$65^{\circ }$
    2. (b)$60^{\circ }$
    3. (c)$50^{\circ }$
    4. (d)$40^{\circ }$
    Show answer

    (c) $50^{\circ }$

    Step-by-step solution →