1. What is an Arc?
An arc is a part of the circumference of a circle. It is like a curved segment joining two points on the circle.
If \(A\) and \(B\) are points on the circle, then the curved path from \(A\) to \(B\) on the circle is an arc.
2. Types of Arcs
Arcs are classified based on their lengths and positions on the circle.
2.1. Minor Arc
The minor arc is the smaller arc connecting two points on the circle.
For points \(A\) and \(B\), the minor arc is usually written as Arc AB or \(\overset{\frown}{AB}\).
2.2. Major Arc
The major arc is the bigger arc connecting the same two points \(A\) and \(B\). It goes around the longer way.
It is usually named using three points, such as Arc ACB.
2.3. Semicircle
A semicircle is an arc formed when the endpoints of the arc are opposite ends of a diameter. It represents half the circle.
3. Naming Arcs
Arcs are named using the points on the circle:
- Two-letter naming for minor arcs (e.g., \(\overset{\frown}{AB}\))
- Three-letter naming for major arcs (e.g., Arc ACB)
When multiple arcs exist between the same endpoints, naming becomes important to avoid confusion.
4. Arc Length
The arc length is the measure of how long the arc is along the circle's boundary.
4.1. Formula for Arc Length (Angle in Degrees)
If a central angle \(\theta\) (in degrees) subtends an arc in a circle of radius \(r\), then:
\(\text{Arc Length} = \dfrac{\theta}{360^\circ} \times 2\pi r\)
4.2. Example
In a circle of radius \(7\text{ cm}\), an arc subtends a central angle of \(60^\circ\). The arc length is:
\(L = \dfrac{60}{360} \times 2\pi \times 7 = \dfrac{1}{6} \times 14\pi = \dfrac{14\pi}{6}\text{ cm}\)
5. Arc Measure (in Degrees)
The measure of an arc is equal to the measure of the central angle that subtends it.
If \(\angle AOB = 90^\circ\), then the arc \(AB\) also has a measure of \(90^\circ\).
5.1. Minor vs Major Arc Measure
Minor arc measure is less than \(180^\circ\), while major arc measure is greater than \(180^\circ\).
6. Arcs and Chords
Every arc corresponds to a chord, and every chord defines two arcs.
6.1. Equal Arcs ↔ Equal Chords
If two arcs are equal, their corresponding chords are also equal, and vice versa.
7. Arcs in Real Life
Arcs appear in clocks (curved minute markings), roads with curves, circular tracks, architecture, and many decorative patterns.