Chords of a Circle: The chord of a circle is defined as the line segment joining any two points on the circumference of the circle. It should be noted that the diameter is the longest chord of a circle that passes through the centre of the circle. In this article, we will learn about different components of a circle, with a special focus on chords, their properties and a few other things related to chords.

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What is a Circle?

A circle is the collection of all the points in a plane whose distance from a fixed point is always the same. The fixed point is known as the centre of the circle.

Different Components of a Circle

Centre: The centre of a circle is the point inside the circle from which the distances to the points on the circle’s circumference are equal.

Radius: The radius of the circle is a line segment that joins the centre and any point on the Circumference.

Diameter: The diameter of the circle is the line segment starting from any point on the circumference of the circle, passing through the centre and ending at the point on the Circumference at the opposite side of a circle. The length of a diameter is twice the length of the radius in a circle. The formula for the diameter of the circle: diameter of a circle \(d = 2\,r,\) where \(r\) is the radius of the circle.

Circumference: The boundary of a circle is known as the circumference of the circle.

Semicircle: A semicircle is half of a circle. A diameter divides a circle into two semicircles.

Chord of a Circle

The chord of a circle are discussed below:

Chords of a Circle Definition

The chord is the straight line that joins the two points on the circumference of a circle.
The diameter of a circle is considered the longest chord as it joins two points on the circumference of a circle.

Difference Between Chord and Diameter

Chord: The chord is the line segment that joins any two points on a circle.

Diameter: A chord of the circle that passes through the centre of a circle is known as the diameter of the circle.

All the diameters of a circle are chords, but all the chords are not diameters of the circle.

The lengths of the chords of a circle may differ, but the length of the diameter of a circle is fixed for a particular circle.

The Segment of a Circle

The chord of a circle divides the circular region of a circle into two parts. Each of the parts is known as the segment of the circle.

If the chord is the diameter, then it divides the circular region into two equal parts. However, if the chord is not a diameter, it divides the region into two unequal parts. The bigger region contains the centre of a circle. This region is called the Major Segment, and the smaller region is called the Minor Segment of the circle.

Arc of a Circle

Any part of the circumference of the circle is known as an arc of the circle. If the length of an arc of the circle is greater than a semicircle, it is known as the Major Arc, and if the length of an arc of the circle is smaller than a semicircle, it is known as the Minor Arc. The sum of lengths of the major arc and the minor arc will always give the circumference of the circle.

Properties of Chord

1. If the radius of the circle is perpendicular to the chord, then it bisects the chord.

2. As the distance of the perpendicular from the centre of the circle to the chord increases, the length of the chord decreases and vice versa.
3. Diameter is the longest chord of the circle, as the perpendicular distance from the centre of the circle to a chord is zero.
4. The triangle formed by the chord and the two radii from the ends of the chord to the centre of the circle is an isosceles triangle.

5. The two chords are equal in length if they are equidistant from the centre of the circle. Example: chord \(PQ = RS,\) if \(TU\) is equal to \(UV.\)

Theorems on Chord of a Circle

Theorem 1: The equal chords of the circle subtend equal angles at the centre.
If \(AB\) and \(CD\) are two chords of a circle with centre \(O,\) such that \(AB = CD.\) Then, \(\angle AOB = \angle COD.\)

Theorem 2: If the angles subtended by the chords of a circle at the centre are equal, the chords are equal.
If \(\angle AOB = \angle COD\) then, \(AB = CD.\)

Theorem 3: A perpendicular from the centre of the circle to the chord bisects the chord.

Theorem 4: The line drawn through the centre of the circle to bisect the chord is perpendicular to the chord.
If \(AM = BM,\) then, \(\angle OMA = \angle OMB = {90^{\rm{o}}}\)

Theorem 5: Angle subtended by a chord at the centre is double the angle by it at any point on the circumference.
Here, \(\angle AOB = 2\angle APB\)

Theorem 6: Angles subtended by a chord at different points on the same side of the Circumference are the same in measure.
Here, \(\angle APB = \angle AQB\)

Theorem 7: Sum of the angles subtended by any chord on any points on the major and minor arc is \({180^{\rm{o}}}.\)
Here, \(\angle CPD + \angle CQD = {180^{\rm{o}}}\)

Theorem 8: The angle subtended by the biggest chord (diameter) on the circumference of the circle is a right angle.
If \(AB\) is a diameter, then \(\angle APB = {90^{\rm{o}}}\)

Formula for Chord Length

The two basic formulas for finding the length of a chord of the circle are given below:

1. Chord length using perpendicular distance from the centre of the circle is \({C_{{\rm{len}}}} = 2 \times \sqrt {{r^2} – {p^2}} ,\) where \(p\) is the perpendicular distance from the centre of the circle to the chord.
2. Chord length using trigonometry with angle \(\theta :{C_{{\rm{len}}}} = 2 \times r \times \sin \left( {\frac{\theta }{2}} \right)\)

Solved Examples on Chords of a Circle

Q.1. Identify the length of the chord of a circle with radius \(7\,{\rm{cm}}.\) Also, the perpendicular distance from the chord to the circle is \(4\,{\rm{cm}}.\) Use chord length formula.
Ans: The given parameters are as follows:
Radius \(r = 7\,{\rm{cm}}\)
The perpendicular distance from the centre to a chord, \(d = 4\,{\rm{cm}}\)
Now, use the formula for the chord length as given below:
\({C_{{\rm{len}}}} = 2 \times \sqrt {\left( {{r^2} – {d^2}} \right)} \)
\({C_{{\rm{len}}}} = 2 \times \sqrt {\left( {{7^2} – {4^2}} \right)} \)
\( = 2 \times \sqrt {\left( {49 – 16} \right)} \)
\( = 2 \times 5.744\)
\( = 11.48\)
Hence, the length of the chord will be \(11.48\,{\rm{cm}}\) approximately.

Q.2. The radius of a circle is \(14\,{\rm{cm}},\) and the perpendicular distance from the chord to the centre is \(8\,{\rm{cm}}{\rm{.}}\) Identify the length of a chord.
Ans: Given radius, \(r = 14\,{\rm{cm}}\)
The perpendicular distance \(p = 8\,{\rm{cm}}\)
Now, the formula. Length of the chord \( = 2\sqrt {\left( {{r^2} – {p^2}} \right)} \)
So, the length of the chord \( = 2\sqrt {\left( {{{14}^2} – {8^2}} \right)} \)
\( = 2\sqrt {\left( {198 – 64} \right)} \)
\( = 2\sqrt {\left( {132} \right)} \)
\( = 2 \times 11.5\)
\( = 23\)
Hence, the length of a chord is \(23\,{\rm{cm}}\) approximately.

Q.3. The perpendicular distance from the centre of the circle to a chord is \(8\,{\rm{m}}.\) Calculate the chord length if a circle’s diameter is \(34\,{\rm{m}}.\)
Ans: Distance is \(p = 8\,{\rm{m}}.\)
The diameter is \(D = 34\,{\rm{m}}.\) So the radius is \(r = \frac{D}{2} = \frac{{34}}{2} = 17\,{\rm{m}}.\)
The length of a chord \( = 2\sqrt {\left( {{r^2} – {p^2}} \right)} \)
The length of chord \( = 2\sqrt {\left( {{{17}^2} – {8^2}} \right)} \)
\( = 2\sqrt {\left( {289 – 64} \right)} \)
\( = 2\sqrt {\left( {225} \right)} \)
\( = 2 \times 15\)
\( = 30\)
Thus, the length of the chord is \(30\,{\rm{m}}{\rm{.}}\)

Q.4. The length of the chord of a circle is \(40\) inches. Assume the perpendicular distance from the centre to a chord is \(15\) inches. What is the radius of the chord?
Ans: Length of the chord \(40\) inches.
Distance is \(p = 15\) inches
Radius, \(r = ?\)
Use the formula. Length of the chord \( = 2\sqrt {\left( {{r^2} – {p^2}} \right)} \)
\(40 = 2\sqrt {\left( {{r^2} – {{15}^2}} \right)} \)
\( \Rightarrow 40 = 2\sqrt {\left( {{r^2} – 225} \right)} \)
Square both sides
\(1600 = 4\left( {{r^2} – 225} \right)\)
\( \Rightarrow 1600 = 4\,{r^2} – 900\)
Add \(900\) on both sides.
\(2500 = 4\,{r^2}\)
Dividing both the sides by \(4,\) we get,
\({r^2} = 625\)
\(\sqrt {{r^2}} = 625\)
\(r = – 25\) or \(25\)
Length can never be the negative number, so it is \(25.\)
Hence, the radius of the circle is \(25\) inches.

Q.5. Radius of a circle shown below is \(10\) yards, and the length of \(PQ\) is \(16\) yards. Calculate the distance of \(OM.\)

Ans: \(PQ = \) length of the chord \( = 16\) yards.
Radius, \(r = 10\) yards,
\(OM = \) distance, \(p = ?\)
Length of the chord \( = 2\sqrt {\left( {{r^2} – {p^2}} \right)} \)
\( \Rightarrow 16 = 2\sqrt {\left( {{{10}^2} – {p^2}} \right)} \)
\( \Rightarrow 16 = 2\sqrt {\left( {100 – {p^2}} \right)} \)
Square both the sides
\(256 = 4\left( {100 – {p^2}} \right)\)
\( \Rightarrow 256 = 400 = 4\,{p^2}\)
Subtract \(400\) on both sides.
\( – 144 = – 4\,{p^2}\)
Divide both sides by \( – 4.\)
\(36 = {p^2}\)
\(p = – 6\) or \(6\)
Length can never be the negative number, so it is \(6.\)
Hence, the perpendicular distance is \(6\) yards.

Summary

In this article, few basics terms of a circle, such as its definition, centre, diameter, radius, semicircle etc., are covered. More details about the chord of a circle, such as its definition, properties, uses, theorems, are discussed. By reading this article, the students will have a fair understanding of the chords of a circle.­­­­­­­­

Frequently Asked Questions (FAQs) – Chords of a Circle

Q.1. What is the measure of an angle on a diameter?
Ans: The measure of the angle of diameter which it makes on any point in circumference is \({90^{\rm{o}}}.\)

Q.2. What is the chord of a circle? Show with a diagram.
Ans: A line segment joining any two points of a circle is called the chord of the circle.

Q.3. Which line is known as a chord?
Ans: The chord of a circle is the straight-line segment whose endpoints both lie on the circular arc.

Q.4. What is the relation between the angles subtended by a chord at the different part on the same side of the chord?
Ans: The angles subtended by a chord at the different parts on the same side of the chord are equal in measure.

Q.5. How many chords are there in a circle?
Ans: There are infinite points on the circle, so you can draw an infinite number of chords in the circle.