Circles are geometric shapes you can see all around you. The significance of the concept of a circle can be well understood from the fact that the wheel is one of the ground-breaking inventions in the history of mankind.

**Parts of a Circle**

Circles are
geometric shapes you can see all around you. The significance of the concept of
a circle can be well understood from the fact that the wheel is one of the ground-breaking
inventions in the history of mankind.

A circle, you can describe, is the set of all points
in a plane at a constant distance from a fixed point. The fixed point is the centre of the circle; the constant distance corresponds
to a radius of the circle.

A line that
cuts the circle in two points is called a secant
of the circle.

A line segment
whose end points lie on the circle is called a chord
of the circle.

**Note**

A circle notably differs from a polygon. A polygon (for example,
a quadrilateral) has edges and corners while, a circle is a ‘smooth’ curve.

A chord of
a circle that has the centre is called a diameter
of the circle. The circumference of a circle is its boundary. (We use the term perimeter in the
case of polygons).

In Fig.4.44,
we see that all the line segments meet at two points on the circle.

These line
segments are called the chords of the circle. So, a line segment joining any two
points on the circle is called a chord of the circle. In this figure *AB*,*
PQ *and* RS *are the chords of the circle.

Now place
four points *P*, *R*, *Q* and *S* on the same circle (Fig.4.45),
then *PRQ* and *QSP* are the continuous parts (sections) of the circle.
These parts (sections) are to be denoted by and or simply by and *.*

This continuous
part of a circle is called an arc of the circle. Usually the arcs are denoted in
anti-clockwise direction.

Now consider
the points *P* and *Q* in the circle (Fig.4.45). It divides the whole
circle into two parts. One is longer and another is shorter. The longer one is called
major arc and shorter one is called minor arc .

Now in (Fig.4.46),
consider the region which is surrounded by the chord *PQ* and major arc .
This is called the major segment of the circle. In the same way, the segment containing
the minor arc and the same chord is called the minor segment.

In (Fig.4.47), if two arcs and of a circle subtend the same angle at the centre, they are said to be congruent
arcs and we write,

implies ∠*AOB* = ∠*COD*

**Note**

A diameter of a circle is:

**• **the line segment which
bisects the** **circle.

**• **the largest chord of a
circle.

**• **a line of symmetry for
the circle.

**• **twice in length of a radius
in a** **circle.

Now, let
us observe (Fig.4.48). Is there any special name for the region surrounded by two
radii and arc? Yes, its name is sector. Like segment, we find that the minor arc
corresponds to the minor sector and the major arc corresponds to the major sector.

Circles with
the same centre but different radii are said to be concentric.

Here are
some real-life examples:

Two circles
are congruent if they are copies of one another or identical. That is, they have
the same size. Here are some real life examples:

Consider
a circle in a plane (Fig.4.51). Consider any point *P* on the circle. If the
distance from the centre *O* to the point *P* is *OP,* then

(i) *OP *= radius (If the point* P *lies
on the circle)

(ii) *OP *< radius (If the point* P *Point
lies inside the circle)

(iii) *OP*
> radius(If the point P lies outside the circle)

So, a circle
divides the plane on which it lies into three parts.

**Progress Check**

**Say True or False**

1. Every chord of a circle contains exactly two points of the circle.

2. All radii of a circle are of same length.

3. Every radius of a circle is a chord.

4. Every chord of a circle is a diameter.

5. Every diameter of a circle is a chord.

6. There can be any number of diameters for a circle.

7. Two diameters cannot have the same end-point.

8. A circle divides the plane into three disjoint parts.

9. A circle can be partitioned into a major arc and a minor arc.

10. The distance from the centre of a circle to the circumference
is that of a diameter

1. How many sides does a circle have ?

2. Is circle, a polygon?

** **

We have already
learnt that there is one and only one line passing through two points. In the same
way, we are going to see how many circles can be drawn through a given point, and
through two given points. We see that in both cases there can be infinite number
of circles passing through a given point *P* (Fig.4.52) , and through two given
points *A* and *B* (Fig.4.53).

Now consider
three collinear points *A*, *B* and *C* (Fig.4.14). Can we draw a
circle passing through these three points? Think over it. If the points are collinear,
we can’t?

If the three
points are non collinear, they form a triangle (Fig.4.55). Recall the construction
of the circumcentre. The intersecting point of the perpendicular bisector of the
sides is the circumcentre and the circle is circumcircle.

Therefore
from this we know that, there is a unique circle which passes through *A*,
*B* and *C*. Now, the above statement leads to a result as follows.

**Theorem 6 **There is one and only one circle passing through three non-collinear
points.

Tags : Geometry | Maths , 9th Maths : UNIT 4 : Geometry

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9th Maths : UNIT 4 : Geometry : Parts of a Circle | Geometry | Maths

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