What is the Relationship Between Arcs and Chords?

Step-by-step Guide: Arcs and Chords

Definitions:

• Arc: An arc is a continuous segment of a circle’s circumference.
• Chord: A chord is a straight line segment whose endpoints lie on the circle. Note: The diameter is the longest chord of a circle.

Properties of Chords and Arcs:

• Chords that are equidistant from the center of a circle are equal in length.
• Equal chords of a circle subtend equal angles at the center.
• The perpendicular bisector of a chord passes through the circle’s center.

Relationship between Chords and Arcs:

• Equal chords intercept equal arcs.
• The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

Calculating Length of Chords:
If we know the arc and radius, the chord’s length, \(c\), can be approximated using:
\(
c \approx 2r \sin\left(\frac{\text{arc angle in radians}}{2}\right)
\)

Examples

Example 1:
Two chords, \(AB\) and \(CD\), of a circle are of equal length. If the arc intercepted by chord \(AB\) measures \(80^\circ\), what is the measure of the arc intercepted by chord \(CD\)?
Solution: Given that equal chords intercept equal arcs, the arc intercepted by chord \(CD\) will also measure \(80^\circ\).

Example 2:
In a circle, an arc intercepts an angle of \(40^\circ\) at the boundary. What is the angle subtended by this arc at the center?
Solution: The angle subtended by an arc at the center is double the angle subtended on the boundary. Therefore, the central angle is \(2 \times 40^\circ = 80^\circ\).

Example 1:
Find the length of a chord that intercepts an arc of \(60^\circ\) in a circle with a radius of \(10 \text{ cm}\).
Solution: Convert the angle to radians: \(60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}\). Using the formula:
\(
c \approx 2 \times 10 \times \sin\left(\frac{\pi}{6}\right) \approx 10 \sqrt{3} \text{ cm}
\)

Practice Questions:

1. In a circle with a radius of \(7 \text{ cm}\), what is the approximate length of a chord that intercepts an arc of \(90^\circ\)?
2. If an arc subtends an angle of \(30^\circ\) at the boundary of a circle, what angle does it subtend at the center?

Answers:

1. \( c \approx 2 \times 7 \times \sin(45^\circ) \approx 9.9 \text{ cm}\)
2. \( 2 \times 30^\circ = 60^\circ \)