# NCERT Solutions for Class 10 Maths Chapter 10 Circles

Written by Team Trustudies
Updated at 2021-05-07

## NCERT solutions for class 10 Maths Chapter 10 Circles Exercise - 10.1

Q1 ) How many tangents can a circle have ?

NCERT Solutions for Class 10 Maths Chapter 10 Circles

A circle can have an infinite number of tangents.

Q2 ) Fill in the blanks :
(i) A tangent to a circle intersects it in _________ point(s).

(ii) A line intersecting a circle in two points is called a _______.

(iii) A circle can have _______ parallel tangents at the most.

(iv) The common point of a tangent to a circle and the circle is called ______.

NCERT Solutions for Class 10 Maths Chapter 10 Circles

(i) exactly one

(ii) secant

(iii) two

(iv) point of contact

Q3 ) 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 of PQ is :
(A) $12$ cm

(B) $13$ cm

(C) $8.5$ cm

(D) $\sqrt{119}$ cm

NCERT Solutions for Class 10 Maths Chapter 10 Circles

cm

So, (D) option is correct

Q4 ) Draw a circle and two lines parallel to a given line such that one is tangent and the other, a secant to the circle.

NCERT Solutions for Class 10 Maths Chapter 10 Circles

## NCERT solutions for class 10 Maths Chapter 10 Circles Exercise - 10.2

Q1 ) 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
(A) $7$ cm

(B) $12$ cm

(C) $15$ cm

(D) $24.5$ cm

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Since QT is a tangent to the circle at T and OT is radius,

It is given that OQ = 25 cm and QT = 24 cm,

$?$ By Pythagoras theorem, we have

$?$ radius of the circle is 7 cm

$?$ Option (A) is correct.

Q2 ) In figure, if TP and TQ are the two tangents to a circle with centre O so that , then is equal to
(A) $60°$
(B) $70°$
(C) $80°$
(D) $90°$

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Since TP and TQ are tangents to a circle with centre O so that ,

$?$ and

and

In the quadrilateral TPOQ, we have

[ \therefore Sum of all angles in a quadrilateral = 360° ]

$?$ Option (B) is correct.

Q3 ) If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then is equal to
(A) $50°$
(B) $60°$
(C) $70°$
(D) $80°$

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Since PA and PB are tangents to a circle with centre O,

, and

and

In the quadrilateral PAOB, we have

In and $OBP$, we have

[Common]

[Each = 90°]

$?$ ( By SAS Criterion)

[C.P.C.T.]

$?$

? The (A) is the correct option.

Q4 ) Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Let PQ be a diameter of the given circle with centre O.

Let AB and CD be the tangents drawn to the circle at the end points of the diameter PQ respectively.

$?$ Tangent at a point to a circle is perpendicular to the radius through the point,

$?$ and

[$?$ and are alternate angles]

Hence, the tangents drawn at the ends of a diameter of a circle are parallel.

Q5 ) Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Let AB be the tangent drawn at the point P on the circle with O.

If possible, let PQ be perpendicular to AB, not passing through O. Join OP.

$?$ Tangent at a point to a circle is perpendicular to the radius through the point,

$?$

(Construction)

, which is not possible.

$?$ It contradicts our supposition.

Hence, the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Q6 ) 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.

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Since tangent to a circle is perpendicular to the radius through the point of contact,

$?$

In , we have

$?$ Radius of the circle is 3 cm.

Q7. 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.

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Let O be the common centre of two concentric circles, and let AB be a chord of the larger circle touching the smaller circle at P. Join OP.

? OP is the radius of the smaller circle and AB is tangent to this circle at P,

? .
We know that, the perpendicular drawn from the centre of a circle to any chord of the circle bisects the chord.

So, and .

In , we have

Now, ???? [? ]

? The length of the chord of the larger circle which touches the smaller circle is 8 cm.

Q8 ) A quadrilateral ABCD is drawn to circumscribe a circle (see figure). Prove that AB + CD = AD + BC.

NCERT Solutions for Class 10 Maths Chapter 10 Circles

? Lengths of two tangents drawn from an external point of circle are equal,

, , , and

Adding these all, we get

Hence proved

Q9 ) In figure, 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 .

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Since tangents drawn from an external point to a circle are equal,

$?$ .
Thus in and , we have

[Common]
[Radii of the same circle]
[ By SSS-criterion of congruence]

Similarly, we can prove that

Now,
[Sum of the interior angles on the same side of transversal is 180°]

[$?$ ]

Q10 ) 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 center.

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Let PA and PB be two tangents drawn from an external point P to a circle with centre O. We have to prove that-

In right and $OBP$, we have

[Tangents drawn from an external point are equal]

[Radii of the same circle]

[Common]

[by SSS - criterion of congruence]

and,

and,

But,
[ is right triangle]

Hence proved 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.

Q11 ) Prove that the parallelogram circumscribing a circle is a rhombus.

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Let ABCD be a parallelogram such that its sides touch a circle with centre O.

We know that the tangents to a circle from an exterior point are equal in length.

, , , and

Adding these all, we get

[$?$ ABCD is a parallelogram ]
and ]

$?$

$?$ ABCD is a rhombus.

Q12 ) 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 figure). Find the sides AB and AC.

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Let us join AO, OC, and OB.

It is given that BD = 8cm, CD = 6 cm.

$?$ Lengths of two tangents drawn from an external point of circle are equal.

$?$

cm.

Then, the sides of the triangle are 14 cm, (x+6) cm and (x+8) cm.

,

and,

$?$ Area of

Also, Area of

Squaring, we get

But x cannot be negative,

Thus, AB = x + 8 = 7 + 8 = 15 cm and AC = x + 6 = 6 + 7 = 13 cm.

Hence, the sides AB and AC are 15 cm and 13 cm respectively.

Q13 ) Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

NCERT Solutions for Class 10 Maths Chapter 10 Circles

Let a circle with centre O touch the sides AB, BC, CD and DA of a quadrilateral ABCD at the points P, Q, R and S respectively.

We have to prove that-

and,
Join OP, OQ, OR and OS.

$?$ The two tangents drawn from an external point to a circle subtend equal angles at the centre.

$?$

Now, [$?$ Sum of all the ? s around a point is 360°]

and,

and

Now since $?2+?3=?AOB$ ,

and