To find the length of arc \(BD\) in circle \(C\), we can use the formula for the arc length, which is given by:
\[ L = \frac{\theta}{360^\circ} \cdot C \]
where \(L\) is the arc length, \(\theta\) is the central angle (in degrees), and \(C\) is the circumference of the circle.
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Determine the circumference (C) of the circle:
The circumference \(C\) of a circle is calculated using the formula:\[ C = 2\pi r \]
To find \(r\) (the radius), we will first need to calculate it using the given information about the triangle \(BCD\).
In triangle \(BCD\), we can use the relation:
\[ \text{If } A = r \text{ and } BC = 18, \text{ then } r = \frac{BC}{2 \sin(\frac{m \angle BCD}{2})} \]
Given \(m \angle BCD = 60^\circ\):
\[ r = \frac{18}{2 \sin(30^\circ)} = \frac{18}{2 \cdot \frac{1}{2}} = \frac{18}{1} = 18 \text{ units} \]
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Calculate the circumference of the circle:
\[ C = 2\pi \cdot 18 = 36\pi \]
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Calculate the arc length \(L\):
Given that the angle \(m\angle BCD = 60^\circ\), we can now substitute into the arc length formula:\[ L = \frac{60}{360} \cdot C = \frac{60}{360} \cdot 36\pi = \frac{1}{6} \cdot 36\pi = 6\pi \]
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Calculate \(6\pi\) and round to the nearest hundredth:
\[ 6\pi \approx 6 \cdot 3.14159 \approx 18.84954 \]
Rounding to the nearest hundredth gives:
\[ L \approx 18.85 \text{ units} \]
Therefore, the length of arc \(BD\) is approximately \( \boxed{18.85} \) units.