In circle J with MZHJK = 46 and HJ Round to the nearest hundredth. 20 units, find the length of arc HK. %3D H K J 75°F A D 40 P Type here to search
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.
![### Finding the Length of an Arc in a Circle
**Problem Statement:**
In circle \( J \) with \( m \angle HJK = 46^\circ \) and \( HJ = 20 \) units, find the length of arc \( HK \). Round to the nearest hundredth.
**Diagram Explanation:**
The image shows a circle with center \( J \). Two points \( H \) and \( K \) are marked on the circumference of the circle. The radius \( HJ \) measures 20 units. The central angle \( \angle HJK \) is \( 46^\circ \).
**Solution Steps:**
To find the length of arc \( HK \), we will follow these steps:
1. **Calculate the circumference of the circle:**
The circumference \( C \) of a circle is given by the formula:
\[
C = 2 \pi r
\]
where \( r \) is the radius of the circle.
Given:
\[
r = 20 \, \text{units}
\]
Therefore,
\[
C = 2 \pi \times 20 = 40 \pi \, \text{units}
\]
2. **Calculate the fraction of the circle represented by the angle \( \angle HJK \):**
The angle \( \angle HJK = 46^\circ \) forms a fraction of the entire circle which is \( 360^\circ \). This fraction is calculated as:
\[
\frac{46}{360}
\]
3. **Calculate the length of arc \( HK \):**
The length of an arc \( s \) is given by the product of the fraction of the circle and the circumference \( C \):
\[
s = \left( \frac{46}{360} \right) \times 40\pi
\]
Performing the calculation:
\[
s = \left( \frac{46}{360} \right) \times 40 \pi \approx 5.078 \, \text{units}
\]
Therefore, the length of arc \( HK \) is approximately \( 5.08 \) units when rounded to the nearest hundredth.
**Graph/Diagram Description:**
The diagram includes a circle centered at point \( J \). Two radii \( HJ \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9500080-40a0-4cd0-8ef1-168bcceb21f9%2F1df3c83d-2040-4eb9-bebf-7778d5dd5441%2F16w4ulo_processed.jpeg&w=3840&q=75)
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