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 a

^{2} + h

^{2} = r

^{2} => a

^{2} = r

^{2} - h

^{2} => 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.

**$\angle BAD$ + $\angle BCD$ = 180**^{o}**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 = PT**^{2}