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 Arc GJ
#### Problem Statement:
In circle H with \( m \angle GHJ = 66^\circ \) and \( GH = 10 \) units, find the length of arc GJ. Round to the nearest hundredth.
#### Diagram Explanation:
The diagram shows a circle with center H. Points G and J lie on the circumference of the circle, and lines GH and HJ are radii of the circle. The angle \( \angle GHJ \) measures 66 degrees, and the radius \( GH \) is 10 units.
#### Solution Steps:
1. **Understanding the Given Data:**
- \( m \angle GHJ = 66^\circ \)
- Radius \( GH = 10 \) units
2. **Formula for Arc Length:**
The length of an arc (\( s \)) in a circle can be calculated using the formula:
\[
s = r \theta
\]
where \( r \) is the radius and \( \theta \) is the angle in radians.
3. **Converting Degrees to Radians:**
Since the angle is given in degrees, we need to convert it to radians:
\[
\theta = \frac{66^\circ \times \pi}{180^\circ} = \frac{66\pi}{180} = \frac{11\pi}{30} \text{ radians}
\]
4. **Calculating the Arc Length:**
Using the formula \( s = r \theta \):
\[
s = 10 \times \frac{11\pi}{30} = \frac{110\pi}{30} = \frac{11\pi}{3} \text{ units}
\]
5. **Approximating to Nearest Hundredth:**
Evaluate the expression numerically:
\[
s \approx \frac{11 \times 3.14159}{3} \approx 11.52 \text{ units}
\]
So, the length of arc GJ is approximately \( 11.52 \) units.
### Summary:
Given \( m \angle GHJ = 66^\circ \) and \( GH = 10 \) units, the length of arc GJ, when rounded to the nearest hundredth, is \( 11.52 \) units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1ee66a4-8e77-4cb9-b1b9-9bd40d747386%2Fe5f184f6-4c8b-4a0b-b3f4-03fcae3e2ddf%2Fbzqrfam_processed.png&w=3840&q=75)
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