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Chord of a Circle

Chord of a circle is a line segment inside the circle whose both endpoints lie on the circumference of circle. In other words a line that is drawn inside the circle and whose endpoints are on the boundary, is called a chord. A chord of the circle is shown in the figure below.

Chord of a circle

In above figure, line segment PQ is chord of the circle. Its endpoints are lying on the circumference of circle. The length of chord is always less than the length of corresponding arc, i.e. in above figure, the length of PQ is less than the arc length of PQ.

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Properties of Chords

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  • Perpendicular drawn from center of circle to the chord, bisects it.
  • Line segment joining center of circle to the midpoint of chord is perpendicular to the chord.
  • Chords that are at equal distance from center are equal in length.
  • Equal chords are always situated at equal distance from center.

Calculation of Chord Length

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Length of the chord can be calculated in the following two ways:

Case 1: When angle subtended by the chord at center is given as shown in the given diagram.

Chord Length Image

Chord length = $2 r sin ($$\frac{\theta}{2}$$)$

Where, r = radius of circle

and $\theta$ = angle subtended by chord at center.

Case 2: When perpendicular distance of the chord from center is given.

Chord Length

Since the perpendicular distance of chord from center divides a chord in half, therefore

Chord length = $2 \sqrt{r^{2}-a^{2}}$

Where, $r$ = radius


$a$ = perpendicular distance of chord from center of circle.

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