# What is the Relationship Between Arcs and Chords?

Within the graceful confines of a circle, arcs and chords dance in tandem, establishing relationships and patterns that have intrigued mathematicians for centuries. While arcs define sections of a circle's boundary, chords are the linear segments that connect two points on that boundary. Their interplay defines various principles of circle geometry. This blog post provides a comprehensive exploration of the nuances of arcs and chords and their intricate interrelation.

## 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:

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

**Answers:**

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

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