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 Statement:**
In circle \(D\) with \(m \angle CDE = 122^\circ\) and \(CD = 6\) units, find the length of arc \(CE\). Round to the nearest hundredth.
**Diagram Explanation:**
The provided diagram illustrates a circle with center \(D\). The points \(C\) and \(E\) lie on the circumference of the circle. Segment \(CD\) is a radius of the circle and has a length of 6 units. The angle \(\angle CDE\) formed at the center of the circle by the radii \(CD\) and \(DE\) measures \(122^\circ\).
**Graph/Diagram:**
The diagram shows a circle with three key points:
- \(D\) (center of the circle)
- \(C\) (point on the circumference)
- \(E\) (point on the circumference)
The radius \(CD\) measures 6 units. The angle at the center \( \angle CDE \) between these radii is shown as \(122^\circ\).
**Steps to Solve:**
1. Use the formula for the length of an arc:
\[
\text{Arc Length} = \theta \times r
\]
where \(\theta\) is the angle in radians and \(r\) is the radius.
2. Convert the angle from degrees to radians:
\[
\theta = \frac{122^\circ \times \pi}{180^\circ}
\]
3. Use the given radius \(r = 6\) units.
4. Substitute the values into the formula and calculate the arc length.
5. Round the result to the nearest hundredth.
**Answer:**
Box to enter the answer:
```
Answer: ____________________________________ [Submit Answer]
```
**Attempt: 1 out of 2**
(Ensure students follow the steps to solve the problem accurately and enter their final answer in the provided box, rounding to the nearest hundredth as instructed.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F310efe35-c25d-45ff-bebc-963b7d0a0035%2Fb245c5dc-e8de-415e-81bd-3b9c3ef0432c%2Fvzj6db_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
