Pond Ecosystem: An ecosystem refers to a biological community that is made of different types of organisms. These organisms interact with each other to cater...

Pond Ecosystem

October 2, 2024**Arc of a circle:** A circle is the set of all points in the plane that are a fixed distance called the radius from a fixed point known as the centre. The line segment joining a point on the circle to the centre is called a radius. By the meaning of a circle, any two radii have the same length.

An arc is a portion of a curve. An arc usually refers to a part of a circle. A chord, a central angle or an inscribed angle may bisect a circle into two arcs. The minor of the two arcs is called the minor arc.

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A circle is the locus of a point that moves in a plane so that its distance from a given fixed point in the plane is always constant.

Consider a circle \(C(O, r)\). Let \(A_1, A_2, A_3, A_4, A_5, A_6\) be the points on the circle. Then, the pieces \(A_1 A_2, A_3 A_4, A_5 A_6, A_1 A_3\) etc., are all arcs of the circle \(C(O, r)\).

Let \(A\) and \(B\) be two points on a circle \(C(O, r)\). Clearly, the circle is divided into two sections, each of which is an arc. We denote the arc from \(A\) to \(B\) in the counterclockwise direction by \(AB\) and the arc from \(B\) to \(A\) in the clockwise direction by \(BA\). Note that the points \(A\) and \(B\) lie on both \(AB\) and \(BA\).

We denote the length of an arc by \(l(AB)\).

From the above discussion, we have, for any two points \(P\) and \(Q\) on a circle either \(l(AB) < l(BA)\) or \(l(AB) = l(BA)\) or \(l(AB) > l(BA)\)

If \(l(AB) < l(BA)\), then \(AB\) is called the minor arc and \(BA\) is known as the major arc. Thus, arc \(AB\) will be a minor arc or a major arc according \(l(AB) < l(BA)\) or \(l(AB) > l(BA)\).

In below figure \(∠ POQ\) is a central angle of the circle \(C(O, r)\).

The definition of minor and major arcs of a circle by using the concept of central angles are given below:

The collection of those points of the circle that lie on and inside a central angle is called as the minor arc of a circle.

In other words, a minor arc of a circle is a part of the circle intercepted by a central angle, including the two points of intersection.

In the below figure, \(PQ\) is a minor arc, and \(QP\) is a major arc of the circle.

It is obvious from the above discussion that the length of an arc is closely associated with the central angle determining the arc. The larger the central angle, the larger will be the minor arc. Therefore, we define the degree measure of an arc in the name of the central angle as given below.

The degree measure of an arc \(PQ\) is denoted by \(m(PQ)\).

Thus, if degree measure of an arc \(PQ\) is \(θ°\), we write \(m(PQ) = θ°\)

Clearly, \(m(PQ) + m(QP) = 360°\) or, \(m(PQ) + m(QP) = m[C(O,r)]\)

In this section, we will study the congruence of circles and the congruence of arcs.

**Congruent Circles: **Two circles are said to be congruent if and only if either of them can be superposed on the other to cover it exactly.

From the above definition, two circles are congruent if and only if their radii are equal, i.e. \(C(O, r) = C(O, s)\) if \(r = s\) where \(r\) and \(s\) are the radii of the two circles, respectively.

**Congruent Arcs:** If two arcs of a congruent circle can be superposed on the other to cover it exactly. Then, the two arcs of a circle are congruent. This will happen if the degree measures of the two arcs are the same. Thus, we can say that: Two arcs of a circle are congruent if and only if they have the same degree measures.

If two arcs \(PQ\) and \(RS\) are congruent arcs of a circle \(C(O, r)\), we write \(PQ ≅ RS\).

Thus, \(PQ ≅ RS ⇔ m(PQ) = m(RS) ⇔ ∠ POQ = ∠ ROS\).

In the below section, we will state and prove some results relating to the congruent arcs.

**Theorem 1:** If two arcs of a circle (or congruent circles) are congruent, then corresponding chords are equal.

Given: Arc \(PQ\) of a circle \(C(O, r)\) and arc \(RS\) of another circle \(C(O’, r)\) congruent to \(C(O, r)\) such that \(PQ ≅ RS\).

To prove: \(PQ = RS\)

Construction: Draw line segments \(OP, OQ\), \(O’ R\) and \(O’ S\).

Proof:

Case 1: When \(PQ\) and \(RS\) are minor arcs

In triangles \(OPQ\) and \(O’ RS\), we have

\(OP = OQ = O’ R = O’ S = r\)

\(∠ POQ = ∠ RO’ S\) \([∵ PQ = RS ⇒ m(PQ) = m(RS) ⇒ ∠ POQ = ∠ RO’ S]\)

So, by SAS criterion of congruence, we have

\(△ POQ ≅ △RO’ S\)

\(⇒ PQ = RS\)

Case 2: When \(PQ\) and \(RS\) are major arcs.

If \(PQ, RS\) are major arcs, then \(QP\) and \(SR\) are minor arcs.

So, \(PQ ≅ RS\)

\(⇒ QP ≅ SR\)

\(⇒ QP = SR\)

\(⇒ PQ = RS\)

Hence, \(PQ ≅ RS ⇒ PQ = RS\)

Consider a circle of radius \(r\) having its centre at the point \(O\). Let \(A, B\), and \(C\) be three points on the circle, as shown in the below figure. The area enclosed by the circle is divided into two regions, namely, \(OBA\) and \(OBCA\). These regions are called sectors of the circle. Each of these two sectors has an arc of the circle as a part of its partition. The sector \(OBA\) has arc \(AB\) as a part of its boundary, whereas \(OBCA\) has arc \(ACB\) as a part of its boundary. These sectors are known as minor and major sectors of the circle, as defined below.

**(I) Minor Sector:** If the minor arc of the circle is a part of the boundary, then a sector of a circle is called a minor sector. In the above figure \(OAB\) is the minor sector.

**(ii) Major Sector:** If the major arc of the circle is a part of its dividing line, then a sector of a circle is called the major sector. In the above figure, sector \(OACB\) is the major sector.

**Note:**

1. The sum of the arcs of a circle’s major and minor sectors is equal to the circle’s circumference.

2. The boundary of a sector is made up of an arc of the circle and the two radii.

If the arc subtends an angle of \(θ\) at the centre, then its arc length \(= \frac{\theta }{{180}} \times \pi r\)

Hence the arc length \(l\) of a sector of angle \(θ\) in a circle of radius \(r\) is given by \(l = \frac{\theta }{{360}} \times 2\pi r\).

Also, the area \(A\) of a sector of angle \(θ\) in a circle of radius \(r\) is given by \(A = \frac{\theta }{{360}} \times \pi {r^2}\)

*Q.1. In the below figure, arc \(AB ≅\) arc \(AC\) and \(O\) is the centre of the circle. Prove that \(OA\) is the perpendicular bisector of \(BC\).*

** Ans:** We have, arc \(AB ≅\) arc \(AC\)

\(⇒ AB = AC\) (Chords of congruent arcs are equal)

\(⇒ OA\) is the bisector of \(∠ BAC\)

\(⇒ ∠ OAB = ∠ OAC\)

Thus, in triangles \(AMB\) and \(AMC\), we have

\(AM = AM\) (Common)

\(AB = AC\) (Proved above)

and, \(∠ OAB = ∠ OAC\) (Proved above)

So, by SAS criterion, we have

\(△ AMB ≅ △ AMC\)

\(⇒ BM = CM\) and \(∠ AMB = ∠ AMC\)

But, \(∠ AMB + ∠ AMC = 180°\)

Therefore, \(∠ AMB = ∠ AMC = 90°\)

Hence, \(AM\) or \(OA\) is the perpendicular bisector of \(BC\).

*Q.2. Given that arc, \(AB\) subtends an angle of \(40°\) to the centre of a circle whose radius is \(7\,\rm{cm}\). Calculate the length of an arc \(AB\).*

** Ans:** Given: Radius \(r = 7\,\rm{cm}\)

\(θ = 40°\)

We know that, length of an arc \( = \frac{\theta }{{360}} \times 2\pi r\)

\( = \frac{{40}}{{360}} \times 2 \times 3.14 \times 7\)

\(= 4.884\)

Therefore, the length of an arc \(AB = 4.884\,\rm{cm}\)

*Q.3. The length of an arc is \(35\,\rm{m}\). If the radius of the circle is \(14\,\rm{m}\), find the angle subtended by the arc.*

** Ans:** We know that, length of an arc \( = \frac{\theta }{{360}} \times 2\pi r\)

\( \Rightarrow 35 = \frac{\theta }{{360}} \times 2 \times 3.14 \times 14\)

\( \Rightarrow \theta = \frac{{360 \times 35}}{{2 \times 3.14 \times 14}}\)

\( \Rightarrow \theta = 143.3^\circ \)

Therefore, the angle subtended by the arc is \(143.3^\circ \).

*Q.4. Find the radius of an arc that is \(156\,\rm{cm}\) in length and subtends an angle of \(150\) degrees to the circle’s centre.*

** Ans:** We know that, length of an arc \( = \frac{\theta }{{360}} \times 2\pi r\)

\( \Rightarrow 156 = \frac{150 }{{360}} \times 2 \times 3.14 \times r\)

\( \Rightarrow r = \frac{{360 \times 156}}{{150 \times 2 \times 3.14}}\)

\( \Rightarrow r = 59.62\,\rm{cm}\)

Therefore, the radius of the circle is \(59.62\,\rm{cm}\)

*Q.5. Calculate the radius of a circle whose arc length is \(144\) yards and arc angle is \(3.665\) radians.*

** Ans:** We know that arc length \(l = rθ\)

\( \Rightarrow 144 = 3.665 \times r\)

\( \Rightarrow r = \frac {144}{3.665}\)

\( \Rightarrow r = 39.29\) yards.

Therefore, the radius of a circle is \(39.29\) yards.

In the above article, we have studied the concept of the arc of a circle, length of an arc, central angle, minor arc and major arc, degree measure of an arc, congruent circles and congruent arcs and the sector of a circle and its area. Also, we solved some example problems based on the length of an arc.

*Q.1. What is an arc in a circle?*** Ans:** A continuous piece of a circle is called an arc of a circle.

*Q.2. How do you find the arc of a circle?*

** Ans:** An arc is a portion of a curve. An arc usually refers to a part of a circle. In a circle \(C(O, r)\). Let \(A_1, A_2, A_3, A_4, A_5, A_6\) be the points on the circle. Then, the pieces \(A_1 A_2, A_3 A_4, A_5 A_6, A_1 A_3\) etc, are all arcs of the circle \(C(O, r)\)

*Q.3. What is the formula for arc measure?*

** Ans:** When \(θ\) is measured in degrees, the measure of the length of an arc \( = \frac{\theta }{{360}} \times 2\pi r\)

When \(θ\) is measured in radians, the measure of the length of an arc \(l = rθ\)

*Q.4. How do you describe an arc?*** Ans:** An arc is a portion of a curve. An arc usually refers to a part of a circle. In other words, a continuous piece of a circle is called an arc of a circle.

*Q.5. What is a major arc of a circle?*** Ans:** A major arc of a circle is the collection of points of the circle that lie on or outside a central angle.

*We hope this detailed article on arc of a circle helped you in your studies. if you have any doubts or queries on this topic, feel to ask us in the comment section and we will be more than happy to assist you. *