Question

In circle J with m, angle, H, J, K, equals, 54, degreesm∠HJK=54
∘
and H, J, equals, 12HJ=12 units, find the length of arc, H, K
HK
⌢
. Round to the nearest hundredth.

Answer 1

To find the length of arc \( \overset{\frown}{HK} \) in circle \( J \), we can use the formula for the length of an arc, which is given by:

\[
L = \frac{\theta}{360} \times C
\]

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

First, we need to calculate the circumference \( C \) of the circle. The formula for the circumference of a circle is given by:

\[
C = 2 \pi r
\]

where \( r \) is the radius of the circle. Since we know the distance \( HJ = 12 \) units, this means the radius \( r \) of the circle \( J \) is 12 units.

Now we can calculate the circumference:

\[
C = 2 \pi (12) = 24\pi
\]

Next, we can use the given angle \( m\angle HJK = 54^\circ \) to find the length of arc \( \overset{\frown}{HK} \):

\[
L = \frac{54}{360} \times 24\pi
\]

Calculating the fraction:

\[
\frac{54}{360} = \frac{3}{20}
\]

Now substituting this back into the arc length formula:

\[
L = \frac{3}{20} \times 24\pi
\]

Now, simplify:

\[
L = \frac{72\pi}{20} = \frac{36\pi}{10} = \frac{18\pi}{5}
\]

Now, we can calculate this value, using \( \pi \approx 3.14 \):

\[
L \approx \frac{18 \times 3.14}{5} \approx \frac{56.52}{5} \approx 11.30
\]

Therefore, the length of arc \( \overset{\frown}{HK} \) is approximately:

\[
\boxed{11.30} \text{ units}
\]

Rounded to the nearest hundredth.