Chord and Arc Properties With Theorems: Definition, Explanation, Examples

Chord and Arc Properties with Theorems: In our day-to-day life, we come across many objects which are round in shape, such as dials of many clocks, wheels of a vehicle, bangles, key-rings etc. In a clock, we observe that the second’s hand rapidly goes around the clock’s dial and its tip moves in a round path. This path traced by the tip of the second’s hand is called a circle.

A continuous part of a circle is known as the arc of the circle, and a line segment joining any two points of a circle is called the chord of the circle. In this article, we will learn about the theorems related to chord and arc properties.

Circle and Some Terms Related to Circle

Before we jump in learning the theorems related to chord and arc properties, let us first look at what a circle is and some terms related to it.

Circle

A circle is a collection of points in a plane, each at a constant distance from a fixed point in that plane. A circle is a path of a point that moves in a plane at a constant distance from a fixed point.

The fixed point is called the centre, and the constant distance is called the circle’s radius. The radius of a circle is always positive. The below-given figure shows a circle with the centre as \(O\) and \(r\) as its radius.

Learn About Chords of a Circle

Chord

A straight line joining any two points on the circumference of a circle is called a chord. In the below-given figure, the straight line \(PQ\) is obtained by joining the points \(P\) and \(Q\) lying on the circle’s circumference. Therefore, \(PQ\) is a chord of the circle. In the same way, \(AB\) too is the chord of the circle.

\(AB\) is the longest chord of the circle called the diameter.

Arc

An arc is a part of the circumference of a circle. Let \(A\) and \(B\) be two points on the circumference of a circle, as shown in the figure given below.

On joining \(AB\) the circle’s circumference is divided into two parts, namely \(APB\) and \(AQB\).

These two parts are known as the arcs of the circle. It is clear from the figure that the length of the arc \(APB\) is smaller than the length of the arc \(AQB\). Therefore, arc \(APB\) is the minor arc, and arc \(AQB\) is the major arc.

Minor And Major Arc:

An arc less than one-half of the whole arc of the circle is called the minor arc of the circle, and an arc greater than one-half of the whole arc is called the major arc of the circle.

Axiom of Equal Arcs: If two arcs subtend equal angles at the centres or centre in equal circles or the same circle, they are equal.

In the below-given figure, arc \(AB\) and arc\(PQ\) of two equal circles having equal radii with centre \(O\) and \(O’\) respectively, subtend equal angles at the centre.

Therefore, \(\angle AOB = \angle P{O^\prime }Q\)

Then, arc \(AB = \) arc \(PQ\).

Converse: If equal circles or in the same circle, if two arcs are equal, then they subtend equal angles at the centre

In the below-given figure, arc \(AB\) and arc \(PQ\) of two equal circles having equal radii with centre \(O\) and \(O’\) respectively, are equal.

Here, arc \(AB = \) arc \(PQ\)

Then, \(\angle AOB = \angle P{O^\prime }Q\).

Theorems on Chord and Arc Properties of Circle

Now, let us see the theorems related to the chord and arc properties of a circle.

Theorem 1: In equal circles or the same circle, equal chords cut off equal arcs.

Given: \(AB\) and \(PQ\) are chords of two equal circles with centre \(O\) and \(O’\) respectively, and \(AB = PQ\).

To prove: Arc \(AB = \) arc \(PQ\)

Construction: Join \(OA,\,OB,\,{O^\prime }P\) and \({O^\prime }Q\).

Proof

Theorem 2: In equal circles or the same circle, if two arcs are equal, then their chords are equal.

Given: Arc \(AB\) and arc \(PQ\) of two equal with centre \(O\) and \(O’\) respectively, and arc \(AB = \) arc \(PQ\)

To prove: \(AB = PQ\)

Construction: Join \(OA,\,OB,\,{O^\prime }P\) and \({O^\prime }Q\)

Proof

Solved Examples – Chord and Arc Properties with Theorems

Q.1. In the below-given figure, arc \(AB = \) arc \(CD\). Prove that \(\angle A = \angle B\).

Ans: Given arc \(AB = \) arc \(CD\)
We know that equal arcs of a circle subtend equal angles at the centre of the circle.
Therefore, \(\angle AOB = \angle COD\)
Now, adding \(\angle BOC\) to both the sides, we get,
\( \Rightarrow \angle AOB + \angle BOC = \angle COD + \angle BOC\)
\( \Rightarrow \angle AOC = \angle BOD\)
In \(\Delta AOC\) and \(\Delta BOD\)
\(OA = OB\) (Radii of the same circle)
\(OA = OB\) (Radii of the same circle)
\(\angle AOC = \angle BOD\) (Proved above)
Therefore,
\(\Delta AOC \cong \Delta BOD\) (By S.A.S. rule of congruency)
Thus, \(\angle A = \angle B\) (By C.P.C.T. )

Q.2. In the below-given figure, chords \(AB\) and \(CD\) of a circle are equal. Prove that \(AD = CB\).

Ans: Given chord \(AB = \) chord \(CD\)
We know that if two chords of a circle are equal, then their corresponding arcs are equal.
Therefore, minor arc \(AB = \)minor arc \(CD\)
\( \Rightarrow \) Arc \(AB = \) arc \(CD\)
Now, subtracting the common arc \(BD\) from both the sides, we get,
\( \Rightarrow \) Arc \(AB – \) arc \(BD = \) arc \(CD – \) arc \(BD\)
\( \Rightarrow \) Arc \(AD = \) arc \(CB\)
According to the theorem, In equal circles or the same circle, if two arcs are equal, their chords are equal.
Therefore, chord \(AD = \) chord \(CB\).
Hence, \(AD = CB\)

Q.3. In the below-given figure, \(AB\) is the diameter of a circle with the centre \(O\). If the chord \(AC = \) chord \(AD\), prove that the arc \(BC = \) arc \(DB\).

Ans: Given chord \(AC = \) chord \(AD\).
According to the theorem, equal chords cut off equal arcs in equal circles or the same circle.
Therefore, minor arc \(AC = \) minor arc \(AD\).
Since \(AB\) is the diameter of the given circle, arc \(ACB = \) arc \(ADB = \).
Therefore,
Arc \(ACB – \) arc \(AC = \) arc \(ADB – \) arc \(AD\)
\( \Rightarrow \) arc \(BC = \) arc \(DB\)

Q.4. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the major arc.

Ans:

In \(\Delta OAB\)
\(AB = OB = OA = \) radius
Therefore, \(\Delta OAB\) is an equilateral triangle.
Thus, each interior angle of this triangle will be equal to \({60^{\rm{o}}}\).
Hence, \(\angle AOB = {60^{\rm{o}}}\)
The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
\(\angle ACB = \frac{1}{2}\angle AOB\)
\( \Rightarrow \frac{1}{2} \times {60^{\rm{o}}} = {30^{\rm{o}}}\)
Hence, the required angle is \({30^{\rm{o}}}\).

Q.5. In the below-given figure, \(AOC\) is the diameter of the circle and arc \(AXB = \frac{1}{2}\) arc \(BYC\). Find \(\angle BOC\).

Ans: As arc \(AXB = \frac{1}{2}\) arc \(BYC\),
\(\angle AOB = \frac{1}{2}\angle BOC\, \ldots ({\rm{i}})\)
Since \(AOC\) is the diameter, \(\angle AOB + \angle BOC = {180^{\rm{o}}}\)
\( \Rightarrow \frac{1}{2}\angle BOC + \angle BOC = {180^{\rm{o}}}\)
\( \Rightarrow \frac{3}{2}\angle BOC = {180^{\rm{o}}}\)
\( \Rightarrow \angle BOC = \frac{2}{3} \times {180^{\rm{o}}} = {120^{\rm{o}}}\)
Hence, the value of \(\angle BOC\) is equal to \({120^{\rm{o}}}\).

Summary

In this article, we first had a quick view of the definition of a circle and two main terms related to it, i.e., chord and arc, then we learnt the theorems and their proofs based on chord and arc properties of a circle. We also learnt the axioms and their proofs of the equal arcs. In addition to this, we solved some examples based on the chord and arc properties theorem.

Learn All About Arc of a Circle

Frequently Asked Questions (FAQs)

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We hope this detailed article on chord and arc properties with theorems helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!