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.