CIRCLE
Circle is a round enclosed figure. It is a set of points in a
plane, which are at equal distance from a given point.
A circle is the locus of a point which moves in a plan in
such a way that its distance from a given point always
remains constant.
O
A
Centre Of Circle
The fixed point is called the centre of the circle. In the figure,
O is the centre of circle.
Radius Of Circle
The constant distance between moving point from fixed point
is called radius of circle. In the figure, OA is the radius of
circle.
Positions Related To Circle
To tell the position of a point making circle as a reference
point, we use various terms.
1. Interior of Circle
The area enclosed by the circle is called interior of the circle.
So, a point P is in the interior of the circle, if distance between
P and centre of the circle is less than the radius of circle, i.e.
OP < radius.
O
Radius
4. Circular Disc
The interior of circle and circle itself taken together are called
circular Disc.
Arc Of Circle
A specific part of the circumference of circle is called arc of the
circle.
A continuous piece of a circle is called arc of a circle.
Example
Points A and B a represents the end-points of an arc. So, AB is
an arc of the circle.
A
O
X
Radius
Arc AB can represented as AB
2. Exterior Of Circle
The area outside the circle is called is exterior of circle. So, a
point P is in the exterior of circle, if distance between P and
centre of circle is more than radius of circle, i.e. OP > radius.
P
O
P
B
P
O
3. Circle Itself
The set of points forming the circle is circle itself. So, a point P
is on the circle if the distance between P and centre of the
circle is equal to the radius., i.e. OP = radius.
Radius
Note
Infact, there are two possible arcs, one on the left side of AB
and second on the right side of AB. So, to differentiate
between these two arcs, we take an extra point for
identification, which in this case is x. So, two arcs formed by
points A and B are arc AB and arc AXB.
1. Minor Arc
The smaller arc is called the minor arc. In the figure, arc AB is
the minor arc.
2. Major Arc
The greater arc is called the major arc. In the figure, arc AXB
is the major arc.
Length Of An Arc
Length of an arc is the part of the circumference covered by
the arc.
Degree Measure Of An Arc
If we join end-points of an arc with the centre of the circle,
then an angle is formed at the centre of circle. This angle is
the degree measures of the arc. In other words, central angle
formed by the arc is the degree measure of arc.
A chord divides the circle in two parts and each part is called
segment of the circle.
Major
Segment
Minor
Segment
A
X
O
Example
B
A
45
0
O
1.
Area of circle enclosed by chord AB and minor arc AB is called
minor segment.
B
2.
On joining the end-points of arc AB with the centre of circle,
an angle of 45o is formed. So degree measure of AB is 45o .
Note
There is difference between the terms AB and m( AB) .
AB means arc AB and m( AB) means measure of arc AB.
Congruent Arcs
Two arcs of a circle are congruent, if their lengths are exactly
same to each other. In this case, the degree measures of the
arcs will also be same.
Two arcs of the circle are congruent if and only if they have
the same degree measure.
Chord Of A Circle
A line with end-points on the circle is called chord of the circle
Example
Line AB has its end-points on the circle. So, AB is chord of the
circle.
A chord passing through centre of the circle is called diameter
of the circle. So, diameter is the longest chord of the circle.
Major Segment
Area of circle enclosed by chord AB and major AXB is called
major segment.
Degree Measure Of An Arc
Degree measure of an arc is the angle made by the end-points
of the arc at the centre of circle. In other hand, degree
measure of an arc is the central angle made by that arc.
In addition, if we join end-points of an arc to any other point
on the circle, then another angle is formed. Now, we will study
the relation between angle made by on arc at centre of circle
and at any other point on the circle.
Cyclic Quadrilateral
If a quadrilateral is such that all its vertices are on the same
circle, then the quadrilateral is called cyclic quadrilateral. In
other words, a quadrilateral is called cyclic quadrilateral if all
its vertices lie on a circle.
C
A
B
C
A
A
O
O
O
C
B
D
B
D
D
Axioms And Theorem
1.
A
O
X
2.
Like an arc, endpoints of chord also form an angle at the
center of the circle. The angle made by end-point of a chord at
the centre is called central angle.
Note
1. By joining the end-points of on arc a chord is formed. So,
an arc makes a chord.
2. A chord divides the circumference of circle in two arc, i.e.
minor arc and major arc. So, a chord makes two arcs.
Equal arcs of a circle subtend equal angle at
centre.
Converse Of Theorem
If angles subtended by two arcs at the centre are
equal, then arcs are equal.
B
Segment Of Circle
Minor Segment
Equal chords of a circle subtend equal angles at the
centre.
Converse Of Theorem
If angles subtended by two chords of a circle at
centre are equal, then the chords are equal.
3.
If two arcs of circle are congruent, then their
corresponding chords are equal.
Converse Of Theorem
4.
If two chords of circle are equal, then their
corresponding arcs are congruent.
Equal chords of circle as equidistant from centre.
Converse Of Theorem
Chords of circle which are equidistant from centre
are equal.
5.
The perpendicular from centre of circle to chord
bisects the chord.
Converse Of Theorem
A line joining the centre of circle to mid-point of
chord is perpendicular to chord.
Perpendicular bisector of chord passes through
center of circle.
6.
Angles in the same segment of a circle are equal.
Note
A segment intercepts an arc, so same theorem can be
stated as an arc.
Angles inscribe by the same arc at the circle are
equal.
Converse Of Theorem
If a line segment joining two points subtended
equal angles at two other points lying on the same
side of line segment, then four points lie on same
circle, i.e. four points are concyclic.
7.
The angle subtended by an arc of a circle at the
centre is double the angle subtended by it at any
point on the remaining part of the circle.
Note
Because, angle subtended by an arc at the centre is also
the degree measure of an arc. So, same theorem can be
stated as degree measure of an arc.
The degree measure of an arc of a circle is twice
the angle subtended by it at any point on the
alternate of the circle with respect to the arc.
8.
Angles in a semi-circle is a right angle.
Converse Of Theorem
Arc of a circle subtending a right angle at any point
on circle has its alternative segment a semi-circle.
9.
The sum of opposite angles of a cyclic quadrilateral
is 1800, i.e. supplementary.
Converse Of Theorem
If sum of any pair of opposite angles of a
quadrilateral is 1800, then the quadrilateral is
cyclic.