Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

Class 10 : Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

The document Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev is a part of Class 10 category.
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Circle

Exercise 1

Question: 1. How many tangents can a circle have?

Solution: On a given point of the perimeter there can be one and only one tangent. Otherwise there can be infinitely many tangents.

Question: 2. Fill in the blanks :

(i) A tangent to a circle intersects it in point (s).

Solution:- One

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

Solution:- Secant

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

Solution: Two

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

Solution: Point of contact

Question: 3. 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 :

(A) 12 cm (B) 13 cm (C) 8.5 cm (D)Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev cm.

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev
Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

Exercise 2

Question: 1. 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

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

Solution: In Δ OPQ

OP2 = OQ2 – PQ2

Or, OP2 = 252 – 242

Or, OP2 = 625 – 576

Or, OP2 = 49

Or, OP = 7

2. In the following figure, if TP and TQ are the two tangents to a circle with centre O so that angle POQ = 110°, then angle PTQ is equal to

(A) 60° (B) 70° (C) 80° (D) 90°

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev
Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

3. If tangents TQ and TP from a point T to a circle with centre O are inclined to each other at angle of 80°, then angle TOP is equal to

(A) 50° (B) 60° (C) 70° (D) 80°

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev
Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

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

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev
Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

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

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

Solution: In the given figure, QR is diameter and O is centre. OP, SP and RP are lines touching the tangent at the point of contact.

In triangles POS and POR

Line PS > PO and SO

Line PR > PO and RO

So, PS and PR are hypotenuse of respective triangles.

So, it is proved that Angle POR = Right Angle

If PO is increased to reach the other end of the perimeter, then it will be the diameter.

6. The length of a tangent from a point Q at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

Solution: In Δ OPQ

OP2 = OQ2 – PQ2

Or, OP2 = 25 – 16 = 9

Or, OP = 3

7. 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.

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

Solution: OS = 5 cm

OT = 3 cm

ST2 = OS2 – OT2

Or, ST2 = 25 – 9 = 16

Or, ST = 4

So, SP = 8 cm

8. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that

AB + CD = AD + BC

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

Solution: AP = AS (tangents from a point are always equal)

Similarly,

PB = BQ

CQ = CR

DR = DS

AB = AP + PB

BC = BQ + CQ = PB + CQ

CD = CR + DR = CQ + DR

AD = DS + AS = DR + AP

AB + CD = AP + PB + CQ + DR ………….. (i)

AD + BC = DR + AP + PB + CQ …………… (ii)

From equation (i) and (ii) it is clear

AB + CD = AD + BC

9. In the given 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 angle AOB = 90°.

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev
Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

10. 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.

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev
Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

So, it is clear that angle subtended by TP and TQ is supplementary to angle subtended by OP and OQ.

11. Prove that the parallelogram circumscribing a circle is a rhombus.

Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev
Circles - NCERT Solution, Class 10 Mathematics Class 10 Notes | EduRev

As diagonals are bisecting each other, so the given parallelogram is a rhombus.

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