Arcs and Chords

Understanding circles is an essential part of geometry, and two fundamental components of circles are arcs and chords. Arcs and chords help us describe pieces of a circle, making it easier to solve geometric problems. In this detailed guide, we will highlight the definitions, properties, and importance of arcs and chords in circles. We will also explore how they relate to each other and to other elements within a circle. By introducing visual and textual examples, we will simplify these concepts to make them more accessible to learners.

What is a circle?

Before diving into arcs and chords, it's important to remember what a circle is. A circle is a simple closed shape. It's the set of all points in a plane that are a fixed distance from a given point, the center. The distance from the center to any point on the circle is called the radius. Here's a representation of a circle:

In the above illustration the centre of the circle is C and the radius is shown by the red line.

Understanding the ragas

A chord is a straight line whose both ends lie on a circle. The length of a chord is less than the diameter of a circle. If a chord passes through the center of a circle, it is called a diameter. Here is an example:

In this example, the blue line AB is a chord of the circle. It connects two points on the circle.

Properties of chords

Chords of a circle have some interesting properties:

• Chords equidistant from the centre of a circle are equal in length.
• Two chords are congruent if the distances from the center to the chords are the same.
• The perpendicular drawn from the centre of a circle to a chord bisects the chord.

As shown in the above figure, the red line perpendicular to the center bisects the blue chord AB.

Defining the arc

An arc is a part or segment of a circle. Arcs are classified into two types based on their size: minor arc and major arc. A minor arc is smaller than a semicircle, while a major arc is larger than a semicircle. An important thing to note is that any two points on a circle form two arcs, a minor arc and a major arc. Here's what an arc looks like:

In the above illustration, the green curve from A to B is the arc called ACB.

Properties of the arc

Arch also has unique features:

• The length of two arcs is equal if the angles subtended by them at the centre are equal.
• The measure of the minor arc is equal to the measure of the central angle that the arc bisects.
• The measure of a major arc is obtained by subtracting 360 degrees from the measure of the corresponding minor arc.

When measuring arcs, the length of the arc can be calculated if the central angle and radius are known. The formula for finding the arc length L derived from the central angle θ in degrees is given as:

L = (θ / 360) × 2πr

Here, r is the radius of the circle, and π is approximately 3.14159.

Relation between arc and chord

Arcs and chords in circles have a special relationship. A chord can be seen as an opening in an arc. A chord divides a circle into two arcs (minor arc and major arc). The larger the arc, the smaller the chord connecting it, and vice versa.

In this illustration, the blue line is a chord that divides the circle into a green major arc and an orange minor arc.

Examples and exercises

It is important to understand the application of arcs and chords in solving geometry problems. Let's explore some problems:

Example 1

Consider a circle of radius 10 cm. A chord subtends a central angle of 60 degrees. Calculate the length of the arc.

To find the arc length we use the formula:

L = (θ / 360) × 2πr
L = (60 / 360) × 2π × 10
L = (1/6) × 2π × 10
L = (10π/3)
L ≈ 10.47 cm

Therefore, the length of the arc is approximately 10.47 cm.

Example 2

In a circle of radius 8 cm, two equal chords are drawn at a distance of 6 cm from the centre. How long are these chords?

To find the lengths of these chords, we use the Pythagorean theorem. The perpendicular drawn from the center to the chord bisects the chord. We know the radius, the distance from the center to the chord, and need to find half the length of the chord (let's call it a):

Use of Pythagorean theorem:

(c^2) = (a^2) + (6^2)
(8^2) = (a^2) + 36
64 = a^2 + 36
a^2 = 64 - 36
a^2 = 28
a = √28
a ≈ 5.29 cm

Since a represents half the length of the chord, the length of the full chord BC is:

BC ≈ 2 × 5.29 cm ≈ 10.58 cm

Therefore, each chord is approximately 10.58 cm long.

Conclusion

By exploring the principles of arcs and chords, we gain a deeper understanding of the structure and characteristics of circles. Arcs divide circles into flexible segments, while chords bridge set distances between points on a circle. Learners are encouraged to visualise these components and tackle a variety of problems to become comfortable with these concepts.

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