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 CISCE for detail information about ICSE Board Class-10.

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

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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 :**

A **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 :**

A **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.

**Radius:**

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 3:

…………………

……………………

…………………..

#### 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 2:

In the figure In a quad. ……………. Cyclic quad.

#### Question 3:

In the figure PQR is …………. QR || ST.

#### Question 4:

……………………

……………………

…………………..

#### 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.

#### Question 3:

………………………

………………………

……………………….

#### Question 10:

#### In figure P and Q ………….. Find AB

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 3:

…………………..

…………………….

…………………..

#### 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 3:

………………………

……………………….

……………………….

#### 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 3:

……………………

……………………

…………………….

#### 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 3:

………………..

………………..

…………………

#### 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.

(b) In the given figure AB is a ……… your answer.

#### Question 2:

…………………….

……………………

……………………

#### Question 28:

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

(i) A circle of ……….. A and C

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

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