Chord of a Circle: Properties, Formula, Theorems – Mindspark

Chord of a Circle

The chord of a circle is a line segment that joins any two points on the circle’s circumference. In the circle given below with ‘O’ as the centre, PQ represents the chord of the circle since it is joining two points on the circumference of the circle. 

The longest chord passes through the centre O, which is called the diameter of the circle.

Chords of a Circle: Important Points

Given below are some of the important points or properties related to the chords of a circle.

• A chord divides the circle into two regions. The region having a bigger area is a major segment, and the other is a minor segment.
• The chord becomes a secant if it is extended infinitely on both the sides.
• If we draw a perpendicular from the centre to the chord of the circle, it bisects the chord.
• The circle’s diameter is the longest chord.
• An isosceles triangle is formed by the chord and the two radii from the ends of the chord to the centre of the circle.

Chords of a Circle: Formula

We can use 2 formulas to find the length of the circle’s chord:

1. Length of the chord using perpendicular distance from the centre

Proof: 

In the circle given below, radius r is the hypotenuse of the triangle formed. Perpendicular bisector d will be one of the sides of the right-angled triangle. 

As per the property of chords, if the circle’s radius is perpendicular to the chord, it bisects the chord.

Thus half of the chord forms the other side of the right-angled triangle. 

So, By Pythagoras theorem,

(1 / 2 \text { chord })^{2}+d^{2}=r^{2}

Or, 1/2 of Chord length =\sqrt{\left(r^{2}-d^{2}\right)}

Therefore, chord length =2 \times \sqrt{\left(r^{2}-d^{2}\right)}

Here r is the circle’s radius; d is the perpendicular distance from the chord to the circle’s centre.

2. Chord length using trigonometry with angle  = 2 × r × sin(ϕ/2)

Where ϕ is the angle subtended at the centre by the chord.

Chords of a Circle: Theorems

Theorem 1

Equal chords of the circle subtend equal angles at the centre of the circle.

Suppose AB and CD are two chords of the circle above with centre O and AB = CD. Then, ∠AOB = ∠COD.

Theorem 2

If the angles subtended at the centre by the chords are equal, the chords of the circle are equal.

According to the theorem, If ∠AOB = ∠COD then, AB = CD.

Theorem 3 

A perpendicular from the centre of the circle to the chord divides the chord into two equal parts.

Theorem 4

If we draw a line through the centre of the circle to bisect the chord, it is perpendicular to the chord.

If AM=BM, then ∠OMA = ∠OMB= 90°

Theorem 5

Angles subtended by a circle chord at different points on the same side of the circumference are of equal measurements.

Here, ∠APB = ∠AQB

Difference Between Radius and Chords of a Circle

The circle’s radius is a line segment that joins the circle’s centre to any point on the circle. On the other hand, the circle’s chord is a line segment connecting any two points on the circumference of the circle. The circle’s diameter is also a chord that passes through the centre and is equal to two times the radius length. It is also the longest chord of a circle.

Examples

1. In the circle given below with O as the centre of the circle, the length of chord DC is 16 cm. Find the length of DE if OF is the circle’s radius?

We know that the radius is perpendicular to the chord of the circle and is a perpendicular bisector. Therefore, 

DE = (½) × AC 

      = (½) × 16

      = 8 cm

2. In the figure given below, O is the centre of the circle with a radius of 5 cm. Find the length of chord DC if the perpendicular from the centre is 4 cm in length?

Since OE is perpendicular to DC, so △DOE will be a right-angled triangle. 

In △DOE, from Pythagoras theorem, 

\mathrm{OD}^{2}=\mathrm{OE}^{2}+\mathrm{DE}^{2}

Or \mathrm{DE}^{2}=\mathrm{OD}^{2}-\mathrm{OE}^{2}

Substituting the values, 

\mathrm{DE}^{2}=5^{2}-4^{2}

Thus, DE = √9 = 3. 

Therefore, Chord DC = 2 x 3 = 6 cm