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# 03 - Let's Recap - Circles - Class 10 - Maths Class 10 Notes | EduRev

## Class 10 : 03 - Let's Recap - Circles - Class 10 - Maths Class 10 Notes | EduRev

``` Page 1

CIRCLES

Circle: A circle is a collection of all points in a plane which are at a constant distance
from a fixed point.
Some parts of a circle:
Chord:  A line segment joining any two end points on the circle is called a chord.
Diameter is the longest chord.
Secant:  A line which intersects the circle in two distinct points is called secant of the
circle.
Tangent: A line which touches the circle at only one point is called tangent to the
circle.

Three different situations that can arise when a circle and a line are given in a
plane, consider a circle and a line PQ. There can be three possibilities as given:

(i) The line PQ and the circle have no common point. Here,
PQ is called a non-intersecting line with respect to the
circle.

(ii) There are two common points A and B that the line PQ and
the circle have. Here, we call the line PQ a secant of the
circle.

(iii) There is only one point A which is common to the line PQ
and the circle. Here, the line PQ is called a tangent to the
circle.

Page 2

CIRCLES

Circle: A circle is a collection of all points in a plane which are at a constant distance
from a fixed point.
Some parts of a circle:
Chord:  A line segment joining any two end points on the circle is called a chord.
Diameter is the longest chord.
Secant:  A line which intersects the circle in two distinct points is called secant of the
circle.
Tangent: A line which touches the circle at only one point is called tangent to the
circle.

Three different situations that can arise when a circle and a line are given in a
plane, consider a circle and a line PQ. There can be three possibilities as given:

(i) The line PQ and the circle have no common point. Here,
PQ is called a non-intersecting line with respect to the
circle.

(ii) There are two common points A and B that the line PQ and
the circle have. Here, we call the line PQ a secant of the
circle.

(iii) There is only one point A which is common to the line PQ
and the circle. Here, the line PQ is called a tangent to the
circle.

Tangent to the circle:

 A tangent to a circle

 The word ‘tangent’
touch and was introduced
1583.

 In the above figure,
A'

B' is a tangent to the circle
 The tangent to a circle
two end points of its

Important Theorems:

Theorem 1: The tangent at any point of a circle is perpendicular to the
through the point of contact.

circle is a line that intersects the circle at
‘tangent’ comes from the Latin word ‘tangere’,
introduced by the Danish Mathematician Thomas
, there is only one tangent at a point of the circle.
is a tangent to the circle.
circle is a special case of the secant, when
its corresponding chord coincide.
Important Theorems:
The tangent at any point of a circle is perpendicular to the
through the point of contact.
at only one point.
‘tangere’, which means to
Thomas Fineke in
only one tangent at a point of the circle.
when the
The tangent at any point of a circle is perpendicular to the radius
Page 3

CIRCLES

Circle: A circle is a collection of all points in a plane which are at a constant distance
from a fixed point.
Some parts of a circle:
Chord:  A line segment joining any two end points on the circle is called a chord.
Diameter is the longest chord.
Secant:  A line which intersects the circle in two distinct points is called secant of the
circle.
Tangent: A line which touches the circle at only one point is called tangent to the
circle.

Three different situations that can arise when a circle and a line are given in a
plane, consider a circle and a line PQ. There can be three possibilities as given:

(i) The line PQ and the circle have no common point. Here,
PQ is called a non-intersecting line with respect to the
circle.

(ii) There are two common points A and B that the line PQ and
the circle have. Here, we call the line PQ a secant of the
circle.

(iii) There is only one point A which is common to the line PQ
and the circle. Here, the line PQ is called a tangent to the
circle.

Tangent to the circle:

 A tangent to a circle

 The word ‘tangent’
touch and was introduced
1583.

 In the above figure,
A'

B' is a tangent to the circle
 The tangent to a circle
two end points of its

Important Theorems:

Theorem 1: The tangent at any point of a circle is perpendicular to the
through the point of contact.

circle is a line that intersects the circle at
‘tangent’ comes from the Latin word ‘tangere’,
introduced by the Danish Mathematician Thomas
, there is only one tangent at a point of the circle.
is a tangent to the circle.
circle is a special case of the secant, when
its corresponding chord coincide.
Important Theorems:
The tangent at any point of a circle is perpendicular to the
through the point of contact.
at only one point.
‘tangere’, which means to
Thomas Fineke in
only one tangent at a point of the circle.
when the
The tangent at any point of a circle is perpendicular to the radius

Remarks:

1. By theorem mentioned
there can be one and only one tangent.

2. The line containing the radius through the point of contact is also sometimes
called the ‘normal’ to the circle at the point.

Number of Tangents from a Point on a Circle
 Case 1: There is no tangent to a circle passing
circle.

 Case 2: There is one and only one tangent to a
lying on the circle

 Case 3: There are exactly two
outside the circle

mentioned above, we can also conclude that at any point on a circle
there can be one and only one tangent.
The line containing the radius through the point of contact is also sometimes
’ to the circle at the point.
Number of Tangents from a Point on a Circle:
There is no tangent to a circle passing through a point lying inside the
There is one and only one tangent to a circle passing through a point
lying on the circle
There are exactly two tangents to a circle through a point lying

above, we can also conclude that at any point on a circle
The line containing the radius through the point of contact is also sometimes
through a point lying inside the
circle passing through a point
circle through a point lying
Page 4

CIRCLES

Circle: A circle is a collection of all points in a plane which are at a constant distance
from a fixed point.
Some parts of a circle:
Chord:  A line segment joining any two end points on the circle is called a chord.
Diameter is the longest chord.
Secant:  A line which intersects the circle in two distinct points is called secant of the
circle.
Tangent: A line which touches the circle at only one point is called tangent to the
circle.

Three different situations that can arise when a circle and a line are given in a
plane, consider a circle and a line PQ. There can be three possibilities as given:

(i) The line PQ and the circle have no common point. Here,
PQ is called a non-intersecting line with respect to the
circle.

(ii) There are two common points A and B that the line PQ and
the circle have. Here, we call the line PQ a secant of the
circle.

(iii) There is only one point A which is common to the line PQ
and the circle. Here, the line PQ is called a tangent to the
circle.

Tangent to the circle:

 A tangent to a circle

 The word ‘tangent’
touch and was introduced
1583.

 In the above figure,
A'

B' is a tangent to the circle
 The tangent to a circle
two end points of its

Important Theorems:

Theorem 1: The tangent at any point of a circle is perpendicular to the
through the point of contact.

circle is a line that intersects the circle at
‘tangent’ comes from the Latin word ‘tangere’,
introduced by the Danish Mathematician Thomas
, there is only one tangent at a point of the circle.
is a tangent to the circle.
circle is a special case of the secant, when
its corresponding chord coincide.
Important Theorems:
The tangent at any point of a circle is perpendicular to the
through the point of contact.
at only one point.
‘tangere’, which means to
Thomas Fineke in
only one tangent at a point of the circle.
when the
The tangent at any point of a circle is perpendicular to the radius

Remarks:

1. By theorem mentioned
there can be one and only one tangent.

2. The line containing the radius through the point of contact is also sometimes
called the ‘normal’ to the circle at the point.

Number of Tangents from a Point on a Circle
 Case 1: There is no tangent to a circle passing
circle.

 Case 2: There is one and only one tangent to a
lying on the circle

 Case 3: There are exactly two
outside the circle

mentioned above, we can also conclude that at any point on a circle
there can be one and only one tangent.
The line containing the radius through the point of contact is also sometimes
’ to the circle at the point.
Number of Tangents from a Point on a Circle:
There is no tangent to a circle passing through a point lying inside the
There is one and only one tangent to a circle passing through a point
lying on the circle
There are exactly two tangents to a circle through a point lying

above, we can also conclude that at any point on a circle
The line containing the radius through the point of contact is also sometimes
through a point lying inside the
circle passing through a point
circle through a point lying

Note: The length of the segment of the tangent from the external point P and the
point of contact with the circle is called the
to the circle.

Theorem 2:  The lengths of tangents drawn from an external point to a circle are
equal.

From the above diagram,
OQ = OR (Radius of the same circle with centre O)
OP = OP (Common side for both the triangles)
Also, ? OQP = ? ORP
? By RHS congruence condition,
?OQP
Which gives us PQ = PR

Remarks:
1. The theorem can also be proved by using the Pythagoras Theorem as follows:
PQ
2
= OP
2
– OQ
2
= OP

2. Note that ? OPQ =
? QPR, i.e. the centre

The length of the segment of the tangent from the external point P and the
point of contact with the circle is called the length of the tangent
The lengths of tangents drawn from an external point to a circle are

From the above diagram, i.e. in right triangles OQP and ORP,
(Radius of the same circle with centre O)
(Common side for both the triangles)

By RHS congruence condition,
OQP ? ?ORP
Which gives us PQ = PR (Corresponding side of congruent triangles)
The theorem can also be proved by using the Pythagoras Theorem as follows:
= OP
2
– OR
2
= PR
2
(As OQ = OR) which gives PQ = PR.
OPQ = ? OPR. Therefore, OP is the angle bisector of
centre lies on the bisector of the angle between the two tangents.
The length of the segment of the tangent from the external point P and the
length of the tangent from the point P
The lengths of tangents drawn from an external point to a circle are
n right triangles OQP and ORP,
(Corresponding side of congruent triangles)
The theorem can also be proved by using the Pythagoras Theorem as follows:
(As OQ = OR) which gives PQ = PR.
OPR. Therefore, OP is the angle bisector of
lies on the bisector of the angle between the two tangents.
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## Mathematics (Maths) Class 10

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