Question

In circle D with m angle C, D, E, = 68 degrees and C, D = 12 units, find the length of CE ⌢ . Round to the nearest hundredth.

Answer 1

To find the length of the arc \( CE \) in circle \( D \), we can use the formula for the arc length, which is given by:

\[
L = \frac{\theta}{360} \times 2\pi r
\]

where:
- \( L \) is the arc length,
- \( \theta \) is the central angle in degrees,
- \( r \) is the radius of the circle.

In your case:
- The measure of angle \( CDE \) (or \( m \angle CDE \)) is \( 68^\circ \).
- The distance \( CD \) is \( 12 \) units, which represents the radius \( r \) of the circle.

We can now substitute the values into the arc length formula:

1. Find the arc length \( L \):

\[
L = \frac{68}{360} \times 2\pi \times 12
\]

2. Simplify inside the parentheses:

\[
L = \frac{68}{360} \times 24\pi
\]

3. Convert \( \frac{68}{360} \):

\[
\frac{68}{360} = \frac{17}{90}
\]

4. Now calculate:

\[
L = \frac{17}{90} \times 24\pi
\]

\[
L = \frac{408\pi}{90}
\]

5. Simplifying gives:

\[
L = \frac{68\pi}{15}
\]

6. Next, we will compute \( L \) using \( \pi \approx 3.14159 \):

\[
L \approx \frac{68 \times 3.14159}{15}
\]

Calculating this gives:

\[
L \approx \frac{213.62912}{15} \approx 14.242
\]

7. Finally, rounding to the nearest hundredth:

\[
L \approx 14.24
\]

Thus, the length of arc \( CE \) is approximately **14.24 units**.