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. $2 \times 30^\circ = 60^\circ$