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

Crash Course for Class 10 Maths by Let's tute

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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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