What is a circle? It is not some thing which invented like a place or an object. It is a symbolic representation of a figure that was invented for the alphabets and numbers. When we look at the wheels, the Sun, the Moon, pulleys, gears, ball bearings, are all circular in shape. Of course the shape of our Earth is spherical. Its cross section is a circle. Apart from this, if we study about the microbiology, the lens used here is circular. You might be aware of Ferry's wheel, which rotates about the fixed axle. We can easily identify the different parts of a circle from this. In this section let us study about the different parts of a circle.

## Center of a Circle

It is the fixed point about which the point moves.
Properties of Circle:
1. A circle can have only one center.
2. We can draw any number of circles from the center.
3. Circles drawn from the same center but different radii are called as concentric circles.
4. circles with the same center and same radii coincide each other.

It is the (shortest) distance between the center of a circle and a point on the circumference. In the above circle it is denoted as r.

1. We can draw many radii in a circle.
2. All the radii of a circle converge at the center of a circle.
3. As the radius of the circle increases the circumference of a circle also increases.

## Chord of a circle

It is the line segment joining any two points on the circumference of a circle.

Properties of Chord of a Circle:
1. We can draw any number of chords in a circle.
2. The chords can either intersect or parallel.
3. Line joining the center of a circle to the midpoint of the chord is perpendicular to the chord.
4. Perpendicular drawn from the center of the circle to the chord bisects the chord.
5. Equal chords are equidistant from the center.

## Diameter of a Circle

It is the chord of a circle which passes through the center of a circle. It is the longest chord of a circle.
Length of the diameter = 2 x radius = 2 r , where ' r ' denote the radius of the circle.
Properties:
1. We can draw many diameters for a given circle. All the diameters converge at the center.
2. Diameter divides the circle into two half.

## Circle Problems

### Solved Examples

Question 1: Draw a circle of radius 3 cm and mark the center as C. Draw a chord of length 5 cm and name it as EF. How many chords of the same lengths can you draw in this circle ?
Solution:

The following diagram shows the circle with center C , whose radius is 3 cm. EF is the chord drawn whose length is 5 cm.
We can draw infinite number of circles

Question 2: Draw a circle of diameter 8 cm and draw its radius. Draw a pair of parallel chords of length 7 cm. Do the chords lie on the same side of the center or on opposite sides of the center. Measure the perpendicular distance between the parallel chords.
Solution:

The following diagram represents a circle of diameter 8 cm.

The chords PQ = RS = 7cm are the parallel chords.
The two chords lie on the opposite sides of the center.
The distance between the parallel chords is 6.5 cm.
[ the above solution is obtained by constructing the above diagram using compasses and scale ]

Question 3: Two circles of radii 3 cm and 7 cm touch internally.
a. What is the distance between their centers.
b. What will be the distance between the centers when they touch externally.
Explain your explanation with suitable diagram.
Solution:

Radii of the two circles are 3 cm and 7 cm respectively.
Let R1 = 3 cm
R2 = 7 cm
Let us draw the circles as shown below.

When the two circles touch internally, the distance between the centers = R2 - R1
= 7 - 3
= 4 cm
When the two circles touch externally, the distance between the centers = R1 + R2
= 7 + 3
= 10 cm

Question 4: A circle of radius 5 cm has its chord at a distance of 3 cm from the center. Find the length of the chord.
Solution:

The following diagram shows a circle of radius 5 cm.
The distance of the chord from the center is 3 cm.
( i. e )  OP = 5 cm
OM = 3 cm

In the $\Delta OPM$, $\angle OMP$ = 90o ,
Therefore applying Pythagoras Theorem,
OM2 + PM2 = OP2
=>      32 + a2     = 52
=>      9  + a2     = 25
=>             a2     = 25 - 9
= 16
=>                  a = $\sqrt{16}$
= 4 cm
Since the perpendicular from the center to a chord, bisects the chord,
length of the chord            PQ = PM + MQ
= a + a
= 2 a
= 2 ( 4 )
= 8 cm
Therefore, the length of the chord = 8 cm

Question 5: A circle of diameter 26 cm has two parallel chords of lengths 24 cm and 10 cm respectively. Find the distance between the chords.
Solution:

The diameter of the circle = 26 cm

Therefore,    the radius of the circle = $\frac{26}{2}$

= 13 cm

In $\Delta OMP$,  $\angle OMP$ = 90o
Therefore applying Pythagoras theorem,
OM2 + PM2 = OP2
=>  h1 2 + 122  = 132
=>  h1 2 + 144 = 169
=>    h1 2        = 169 - 144
= 25
= 52
=>             h1  = 5 cm
In $\Delta ONS$,  $\angle ONS$ = 90o
Therefore applying Pythagoras theorem,
ON2 + NS2 = OS2
=>  h2 + 52   = 132
=> h2 + 25    = 169
=>           h2 = 169 - 25
= 144
= 122
=>              h2 = 12
When the parallel chords are on either side of the center,
the distance between the chords = h1 + h2
= 5 + 12
= 17 cm
When the parallel chords are on the same side of the center,
the distance between the chords = h- h1
= 12 - 5
= 7 cm