# OP Malhotra Circle Class-10 S.Chand ICSE Maths Solutions Ch-14

OP Malhotra Circle Class-10 S.Chand ICSE Maths Solutions Ch-14.  We Provide Step by Step Answer of Exe-14 with Self Evaluation and Revision of S Chand OP Malhotra Maths . Visit official Website for detail information about ICSE Board Class-10.

## OP Malhotra Circle Class-10 S.Chand ICSE Maths Solutions Ch-14

-: Select Topics :-

Exercise-14(a)

Exercise-14(b)

Exercise-14(c)

Exercise-14(d)

Exercise-14(e)

Exercise-14(f)

Exercise-14(g)

Self Evaluation And Revision

Exercise-14

### OP Malhotra Circle Class-10 S.Chand ICSE Maths Solutions Ch-14

Circle and line in a plane :

For a circle and a line on a plane, there can be three possibilities.

i) they can be non-intersecting

ii) they can have a single common point: in this case, the line touches the circle.

ii) they can have two common points: in this case, the line cuts the circle.

#### Tangent :

tangent to a circle is a line which touches the circle at exactly one point. For every point on the circle, there is a unique tangent passing through it.

Secant :

secant to a circle is a line which has two points in common with the circle. It cuts the circle at two points, forming a chord of the circle.

Circle:

A circle is a collection of all points in a plane which are at a constant distance from a fixed point.

Centre:

The fixed point is called the centre.

The constant distance from the centre is called the radius.

Chord:

A line segment joining any two points on a circle is called a chord.

Diameter:

A chord passing through the centre of the circle is called diameter. It is the longest chord.

Tangent:

When a line meets the circle at one point or two coincidings The line is known as points, a tangent.
The tangent to a circle is perpendicular to the radius through the point of contact.
⇒ OP ⊥ AB

The lengths of the two tangents from an external point to a circle are equal.
⇒ AP = PB

Length of Tangent Segment
PB and PA are normally called the lengths of tangents from outside point P.

Properties of Tangent to Circle

Theorem 1: Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Given: XY is a tangent at point P to the circle with centre O.
To prove: OP ⊥ XY
Construction: Take a point Q on XY other than P and join OQ
Proof: If point Q lies inside the circle, then XY will become a secant and not a tangent to the circle
OQ > OP

This happens with every point on the line XY except the point P. OP is the shortest of all the distances of the point O to the points of XY
OP ⊥ XY …[Shortest side is the perpendicular]

Theorem 2: A line drawn through the end point of a radius and perpendicular to it, is the tangent to the circle.
Given: A circle C(O, r) and a line APB is perpendicular to OP, where OP is the radius.
To prove: AB is tangent at P.
Construction: Take a point Q on the line AB, different from P and join OQ.
Proof: Since OP ⊥ AB
OP < OQ ⇒ OQ > OP

The point Q lies outside the circle.
Therefore, every point on AB, other than P, lies outside the circle.
This shows that AB meets the circle at point P.
Hence, AP is a tangent to the circle at P.

Theorem 3: Prove that the lengths of tangents drawn from an external point to a circle are equal
Given: PT and PS are tangents from an external point P to the circle with centre O.
To prove: PT = PS
Construction: Join O to P, T and S.

Exe-14(a),

### OP Malhotra Circle Class-10 S.Chand ICSE Maths Solutions Ch-14

#### Question 1:

Fill up the blanks :

(i) O is the center of the circle .

………………….

#### Question 2:

In the figure O is the centre of the ……. Step of working.

…………………

……………………

…………………..

#### Question 27:

In the figure AB = AC = CD, …………….. Calculate

(i) angle ABC

(ii) angle BEC

### Exe-14(b),

OP Malhotra Circle Class-10 S.Chand ICSE Maths Solutions Ch-14

#### Question 1:

In the figure ……………. is a cyclic quadrilateral.

#### Question 3:

In the figure PQR is …………. QR || ST.

……………………

……………………

…………………..

#### Question 12:

ABCD is a cyclic ………….. prove taht EF || DC.

### Exe-14(c),

Circle Class-10 S.Chand ICSE Maths Solutions Ch-14

#### Question 1:

(a) In figure …………… angle POQ

(b) In figure ………. fiond AC

(c) IN the figure ……………. angle QOR

#### Question 2:

In figure O is the center of the …………….. Find angle POS.

………………………

………………………

……………………….

Exe-14(d),

### OP Malhotra Circle Class-10 S.Chand ICSE Maths Solutions Ch-14

#### Question 1:

In figure PA and PB are ………………….. KN = AK + BN

#### Question 2:

In figure two ………….. point of contact.

…………………..

…………………….

…………………..

#### Question 12:

PA and PB are tangent drawn ……………… angle APB = 2 angle CAB

### Exe-14(e),

Circle Class-10 S.Chand ICSE Maths Solutions Ch-14

#### Question 1:

In figure if AB and CD are two ………… such that

(i) A P = 8 cm ………….

(ii) AB = 12 cm …………..

(iii) AP = 6 cm ……………..

#### Question 2:

In the figure if AB and CD are two ………… Such that

…………………….

………………………

……………………….

……………………….

#### Question 16:

In figure PB = BT and PT …………. prove that (i)………….. (ii)…………..

Exe-14(f),

### OP Malhotra Circle Class-10 S.Chand ICSE Maths Solutions Ch-14

#### Question 1:

In figure PT is a tangent ……………. find angle ABT.

#### Question 2:

In figure TAS is a ……….. of x and y.

……………………

……………………

…………………….

#### Question 10:

In the figure CD is ………………. triangle DCA

### Exe-14(g),

OP Malhotra Circle Class-10 S.Chand ICSE Maths Solutions Ch-14

#### Question 1:

Using ruler and …………. one of them.

#### Question 2:

Draw a tangent to a circle …………. a given line.

………………..

………………..

…………………

#### Question 16:

Draw a regular ………….. a circle on it.

Self Evaluation And Revision

### OP Malhotra Circle Class-10 S.Chand ICSE Maths Solutions Ch-14

#### Question 1:

(a) In the figure Chord ……….. length of CD.

…………………….

……………………

……………………

#### Question 28:

Draw a line AB = 5 cm …………………… construct

(i) A circle of ……….. A and C

(ii) Construct two tangents to …………… of the tangents.

— : End of Circle OP Malhotra S Chand Solutions :–

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