Chord of a Circle, Its Length and Theorems

Introduction

A chord is a line segment connecting any two points located on the circumference of the circle . The circle is a well-known two-dimensional shape used in Euclidean geometry. It has various components, including chord, radius, diameter, etc. In this tutorial, we will discuss the definition, formulae, and some theorems related to the chord of a circle.

Circles

• The circle is a two-dimensional figure drawn on a plane in such a way that each point on the circle is equidistant to a fixed point.

• The fixed point is known as the center of the circle.

• The dimension of the circle is expressed in terms of its radius.

• The distance between the origin and any point on the circle is called the radius.

• The word “circle” is derived from the Latin word “circulus” (means small ring). The graphical presentation of a circle is depicted below.

There are various properties of a circle, which are briefly summarized below.

• The circle is a curved face, and any straight line bounds the curved region. Hence, it is not included in the polygon category.

• The circles having the same radius are considered congruent circles.

• The perpendicular bisector of any chord passes through the center.

• The longest chord is the diameter of the circle.

• The line segment connecting the intersecting points of two circles is orthogonal to the line segment joining the centers of the two circles.

• The circle can be inscribed in a triangle or square.

• The tangents drawn at the endpoints of the diameter are parallel to each other.

Chord of a Circle

• The chord is the line segment joining any two points on the circle.

• In other words, the endpoints of the chord lie on the circumference of the circle.

• An infinite number of chords can be drawn in a circle.

• The line segments AB, CD, and EF are some chords represented in the following figure.

There are several properties of chords that are summarized below.

• The diameter is the longest chord of the circle.

• A chord divides the circle into two parts.

• The diameter divides the circle into two equal parts.

• When a chord is extended on both sides, it is known as a secant.

Length of a Chord Given its Distance From the Centre

If the distance of the chord from the center is given, the length of the chord can be evaluated by using the following formula.

$$\mathrm{S\:=\:2\sqrt{(r^{2}\:-\:p^{2})}}$$

where r = radius of the circle

p = perpendicular distance of the chord from center

S = length of the chord

Proof −

In the above figure, the hypotenuse of the right-angled triangle is the radius of the circle. The perpendicular bisector is one of the sides of the triangle. It is well known that the perpendicular line bisects the chord. Therefore, the chord length will be twice the base of the right-angled triangle.

Now, using the Pythagorus theorem,

$$\mathrm{Base\:=\:\sqrt{(r^{2}\:-\:p^{2})}}$$

$$\mathrm{\Longrightarrow\:chord\:=\:2\times\:=\:=\:2\sqrt{(r^{2}\:-\:p^{2})}}$$

Length of a Chord Given the Subtended Angle

Let’s consider a circle having a radius of r and a chord length of S. The chord makes an angle of 𝚹 at the center.

Using trigonometry, the length of a chord $\mathrm{=\:2\times\:r\times\:\sin\frac{\theta}{2}}$

Theorems of Chord of a Circle

There are various theorems associated with the chord, which are described below.

• Theorem 1 − The perpendicular line drawn from the center to a chord bisects the chord. In the given figure, TS = SM.

• Theorem 2 − Two chords are said to be equal if the perpendicular lines from the center to the respective chord are equal in length. In the given figure, TM = AB.

• Theorem 3 − If two chords subtend at an equal angle at the center, two chords are equal.

• Theorem 4 − For two unequal chords, the larger one is closer to the center compared to the smaller one

Solved Examples

1)In the given figure, OP = 10 cm, TS = 6 cm. Find the length of the chord TM?

Answer −

The radius of the circle $\mathrm{=\:r\:=\:OP\:=\:10\:cm}$

Base of the right-angled triangle = TS = 6 cm

Using the Pythagorean theorem,

$$\mathrm{OS\:=\:\sqrt{(OP^{2}\:-\:TS^{2})}}$$

$$\mathrm{\Longrightarrow\:OS\:=\:\sqrt{(10^{2}\:-\:6^{2})}}$$

$$\mathrm{\Longrightarrow\:OS\:=\:\sqrt{64}\:=\:8\:cm}$$

Using the formula of the chord,

$$\mathrm{\Longrightarrow\:chord\:=\:2\sqrt{(radius^{2}\:-\:perpendicular^{2})}}$$

$$\mathrm{\Longrightarrow\:chord\:=\:2\sqrt{(10^{2}\:-\:8^{2})}}$$

$$\mathrm{\Longrightarrow\:chord\:=\:12\:cm}$$

∴ The length of the chord is 12 cm.

2)Find the length of the chord and TS if OT = 15 cm and OS = 4 cm

Answer −

In the right-angled triangle OTS,

$$\mathrm{OT^{2}\:=\:OS^{2}\:+\:TS^{2}}$$

$$\mathrm{\Longrightarrow\:15^{2}\:=\:4^{2}\:+\:TS^{2}}$$

$$\mathrm{\Longrightarrow\:TS^{2}\:=\:15^{2}\:-\:4^{2}}$$

$$\mathrm{\Longrightarrow\:TS^{2}\:=\:225\:-\:16\:=\:209}$$

$$\mathrm{\Longrightarrow\:TS\:=\:\sqrt{209}\:=\:14.46\:cm}$$

The length of the chord $\mathrm{=\:TM\:=\:2\times\:TS\:=\:2\times\:14.46\:=\:28.92\:cm}$

∴ The length of the chord and TS are 28.92 and 14.46 cm, respectively.

Word Problems

• Problem 1 − Find the length of a chord if the length of diameter and perpendicular from the center to the chord are 18 and 3 cm, respectively.

• Problem 2 − A chord makes an angle of 30° at the center of a circle. The radius of the circle is 10 cm. Find the length of the chord.

• Problem 3 − A chord of a circle is half of the radius. Find the angle subtended by the chord at center.

Conclusion

The present tutorial gives a brief introduction about the chord and its various properties. In addition, some basic formulae related to the length of the chord have been briefly provided. Some theorems related to the chord of a circle have been stated in this tutorial. Moreover, some solved examples have been provided for better clarity of this concept. In conclusion, the present tutorial may be useful for understanding the basic concept of the chord and its theorems.

FAQs

1. Can a chord be shorter than the radius of the circle?

Yes. It is not necessary that the length of the chord should always be greater than the radius.

2. How many chords can be drawn in a circle?

There can be an infinite number of chords that can be drawn in a circle. However, diameter is the greatest chord of a circle.

3. What is the difference between a chord and a sector?

A chord is a line segment joining any two points on the circumference of the circle, whereas a sector is a part of a circle that is made from two radii and one arc. A circle can be divided into two sectors, whereas an infinite number of chords are drawn in a circle

4. Can a tangent be called a chord?

A tangent is a line segment that intersects the circle at only one point, whereas the chord is a line segment that intersect the circle at two points. Therefore, a tangent cannot be called a chord.

5. Which triangle is formed when two radii join the two ends of a chord?

If two radii join the endpoints of a chord, an isosceles triangle is formed. In addition, the perpendicular line from the center to the chord bisects the chord.