Circles: Parts and Theorems (Based on CBSE Class 10)

1. B
y:
Va
ru
n

2. • Circle – set of all points in a
plane that are equidistant from a
given point called a center of the
circle. A circle with center P is
called “circle P”, or P.
• The distance from the center to a
point on the circle is called the
radius of the circle. Two circles
are congruent if they have the
same radius.

3. O
=
=
=
A
C D
EB
Radius
Diameter
Centre
of circle
Circumference
A circle is a shape with all points at the same distance from its
centre.
It’s the distance across
a circle through the
center.
It’s the distance from
the center of a circle
to any point on the
circle.
All points on the circle are
at same distance from the
centre point.
It’s the distance around
the circle.

4. O
A
EB
Chord Chord
Diameter,
the biggest chord
Interior of
the circle
Exterior of the circle
A chord is a straight line segment with its end points on the circumference
of a circle

5. O
M
N
Chord
Major Arc
An arc is a part of a circumference of a circle.
Minor Arc
1
2

6. A chord divides the circular region into 2 parts, each of which is called
a segment of the circle.
D E
Chord DE
Minor segment of a circle
Major segment of a circle
S
R
 The part of the
circular region
enclosed by a
major arc and
the chord is
called a major
segment.
 Minor
segment
does not
contain the
centre of
the circle.

7. O
A B
Radius Radius
Minor
Sector
Major
Sector
Sector of a circle is a portion of a circle enclosed by two radii and
an arc.

8. O
Secant
Line AB and the circle
have two common
points M and N
Tangent
There is only one point
P which is common to
the line AB and the
circle
A
B
O
M
N
O
P
Non-intersecting Line
Line AB and the
circle have no
common points

10. The circumference of a circle is the
distance around it. The term is used
when measuring physical objects, as
well as when considering abstract
geometric forms.
The area of a circle is the number of square
units inside that circle.
A = pr
C = 2pr
2

11. Q
P
• IF A LINE IS TANGENT TO A
CIRCLE, THEN IT IS
PERPENDICULAR TO THE RADIUS
DRAWN TO THE POINT OF
TANGENCY.
• IF L IS TANGENT TO Q AT POINT P,
THEN L ⊥QP.
• THE TANGENT AT ANY POINT OF A
CIRCLE IS PERPENDICULAR TO THE
RADIUS THROUGH THE POINT OF
l

12. Construction: Draw a circle with centre O.
Draw a tangent XY which touches point P at
the circle.
We need to prove that OP is perpendicular to
XY.
Take a point Q on XY other than P and join
OQ.
The point Q must lie outside the circle.
Therefore, OQ is longer than the radius OP of
the circle. That is,
OQ > OP.
Since this happens for 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.
So OP is perpendicular to XY.
Remarks:
1. By the theorem above, we can also conclude that
at any point on a circle there can be one and only
one tangent
2. The line containing the radius through the point of
contact is also sometimes called ‘normal’ to the
point
Proof:

13. The lengths of tangents drawn
from an external point to a circle are equal.
Proof : We are given a circle with centre O, a
point P lying outside the circle and two tangents
PQ, PR on the circle from P. We
are required to prove that PQ = PR.
For this, we join OP, OQ and OR. Then
angle OQP and angle ORP are right angles, because
these are angles between the radii and tangents,
and according to Theorem 10.1 they are right
angles. Now in right triangles OQP and ORP,
OQ = OR (Radii of the same circle)
OP = OP (Common)
Therefore, OQP ORP (RHS)
This gives PQ = PR (CPCT)angle

14. Remarks :
1.The theorem can also be proved by using the
Pythagoras Theorem as follows:
2PQ = 2OP – 2OQ = 2OP –2OR =2PR (As OQ =
OR)
which gives PQ = PR.
Hence Proved
2. Note also that ∠ OPQ = ∠ OPR. Therefore,
OP is the angle bisector of ∠ QPR, i.e., the
centre lies on the bisector of the angle
between the two tangents.

15. The circle is one of the most
common shapes in our daily life,
and indeed the universe. Planets,
the movement of the planets,
natural cycles, natural shapes -
there are circles absolutely
everywhere. The circle is one of
the most complex shapes,
and indeed the most difficult for
man to create, yet
nature manages to do it perfectly.
The centres of flowers, eyes, and
many more things are circular and
we see them in our every day life.

16. However, circles are abundant in our daily
life in regards to manmade objects, too.
Circles can be found in modern
architecture, in the house and out on the
street. Without circles we wouldn't have
cars. Cars only work because of the circular
motion that is produced to make the circular
wheels go round. Without circles the motion
wouldn't be created and the wheels simply
wouldn't move properly, or make the car
move.
If you're into sports, then a sudden lack of
circles would completely ruin your sporting
activities. In many sports, balls are
necessary (football, tennis, ping pong, golf,
and so many more). Without circles there
would be no spheres, and these sports
simply wouldn't be possible for anybody to
play.