Circle is a closed figure, which is the path traced by a point which moves such that it moves with a constant distance from the fixed point. We are already aware that this constant distance is called the radius. The chord of a circle is the one which divides the circle into two parts called major segment and minor segments. The chords have different properties regarding the angle subtended at the center, area divided inside the chord etc. In this section let us discuss about the properties of the chord of a circle.

## Chord Definition

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

Diameter: If the chord passes through the center of the circle then it is called the Diameter of the circle.

The chord divides the circle into two segments called major segment and minor segment.

The following diagrams describe these parts of the circle.

## Circle Chord Formulas

A. If the chord passes through the center of a circle, length of the chord = length of the diameter = 2 x radius = 2 r

B. If the Chord does not pass through the center of a circle we draw perpendicular from the center and use the following properties to find the length of the chord.

### Chord properties:

Property 1: Perpendicular drawn from the center of a circle to a chord bisects the chord.

Property 2: Line joining the center of a circle to the midpoint of a chord is perpendicular to the chord.

To find the length of the chord of a circle of radius ' r':
Proof:
Let the radius of the circle  = r units
Let the length of the chord be 2 a units
Let the length of the perpendicular from the center of a circle to the chord = h
Since the perpendicular divides the chord into two equal parts, we apply Pythagoras theorem for the right angled triangle shown above.
Therefore a2 + h2 = r2
=>            a2 = r2 - h2
=>             a = $\sqrt{r^{2}-h^{2}}$
Length of the chord of the circle = 2 a = 2 $\sqrt{r^{2}-h^{2}}$

Central Angle: Angle made by the chord at the center is called the central angle.
In the above figure, the central angle is $\angle BOC$
Inscribed Angle: Angle made by the chord at any point on the circumference is called the inscribed angle
In the above figure, the inscribed angle is $\angle BAC$

Property  3: The central angle made by a chord is double the angle made by it any point on the circumference.
In the above figure, $\angle BOC$ = 2 $\angle BAC$
Property 4: The inscribed angles made by a chord at any point on the circumference have equal measure.

In the above figure, $\angle BAC$ =  $\angle BPC$
Property 5: Angle inscribed by a chord in the minor segment is Obtuse and the angle inscribed by a chord in the major segment is acute.

In the above diagram, $\angle BAD$ is in the major segment and hence it is acute.
$\angle BCD$ is in the minor segment and hence it is obtuse.

Property 6: The sum of the angles inscribed by a chord in the major and the minor segments of the circle is supplementary.
$\angle BAD$ +  $\angle BCD$ = 180o
Property 7: Equal chords are equidistant from the center.

In the following diagram, the chords AB and CD are equal in length.
Therefore, their distances are equal.
(i . e) OM = ON

Property 8: IF two chords of a circle intersect internally, then the product of the lengths of their segments is equal.
In the following diagram, the chords AB and CD intersect at P,
Therefore, PA x PB = PC x PD

Property 9: The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment.

In the above diagram, The chord AB makes angle with the tangent PQ at A.
Therefore, $\angle BAQ$ = $\angle BCA$
Similarly   $\angle PAB$ = $\angle ADB$

Property 10: IF a chord and a tangent intersect externally, then the product of the lengths of the segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.

In the above diagram, the chord AB and the tangent aTP of the circle intersect each other at the point P outside the circle.
Therefore, according to the above property, PA x PB = PT2

## How to find the Chord of a Circle

### Solved Examples

Question 1: Find the length of the chord of a circle or radius 10 cm, which is at a distance of 6 cm from the center.

Solution:

We have, radius of the circle = 10 cm
Perpendicular distance from the center = 6 cm
Applying the above formula, a = $\sqrt{r^{2}-h^{2}}$, where r = 10 and h = 6
=>              a = $\sqrt{10^{2}-6^{2}}$
=  $\sqrt{100 - 36}$
= $\sqrt{64}$
= $\sqrt{8^{2}}$
=>          Length of the chord = 2 a
= 2 ( 8 )
= 16 cm

Question 2: Two chords of a circle of radius 5 cm are of length 6 cm. Find the length of each chord.
Solution:

We are given that the radius of the circle = 5 cm
Length of the chord = 2 a
= 6 cm
=>      a = 6/2
= 3 cm
Let h be the distance of the chord from the center.
Therefore using the above formula ( Pythagoras theorem )
we get, a2 + h2 = 52
=>        32 + h2 = 25
=>               h2 = 25 - 9
= 16
=>                h = $\sqrt16$
= 4 cm
Since the two chords are of equal lengths, the distance of each chord from the center = 4 cm.
Note: If the chords are parallel, then
1. The distance between the chords is 4 + 4 = 8 cm ( when the chords lie on either side of the center of the circle ).
2. The distance between the chords  = 4 - 4 = 0 ( when the chords lie on the same side of the center )  (i . e ) the chords coincide each other.