Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![**Problem Description**
In circle \( J \) with \( m \angle HJK = 48^\circ \) and \( HJ = 11 \) units, find the length of arc \( HK \). Round to the nearest hundredth.
**Diagram Explanation**
The image shows a circle centered at point \( J \). Inside the circle, there are two points, \( H \) and \( K \), on the circumference. Line segments \( HJ \) and \( JK \) are drawn, forming an angle \( \angle HJK \) at the center \( J \). The measure of \( \angle HJK \) is given as \( 48^\circ \).
**Solution Steps**
1. **Calculate the Circumference of the Circle:**
The circumference \( C \) of the circle is given by \( C = 2\pi r \).
2. **Find the Length of Arc \( HK \):**
The length of arc \( HK \) can be found as a fraction of the circumference. Since the angle \( \angle HJK \) is \( 48^\circ \), we use the formula for the arc length:
\[
\text{Arc Length} = \left( \frac{\theta}{360} \right) \times 2\pi r
\]
Substituting the given values \( \theta = 48^\circ \) and \( HJ = 11 \) units:
\[
\text{Arc Length} = \left( \frac{48}{360} \right) \times 2\pi \times 11
\]
3. **Simplify and Calculate:**
Simplifying inside the parenthesis first:
\[
\left( \frac{48}{360} \right) = \frac{4}{30} = \frac{2}{15}
\]
Then substitute this into the formula:
\[
\text{Arc Length} = \left( \frac{2}{15} \right) \times 2\pi \times 11
\]
Simplifying further:
\[
\text{Arc Length} = \frac{4\pi \times 11}{15} = \frac{44\pi}{15}
\]
Using \( \pi \approx 3.14 \):
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