To find the length of the chord \( FG \) in circle \( C \) given that the radius \( r = 8 \) mm and the angle \( m\angle FCE = 60^\circ \), you can use the formula for the length of a chord:
\[ FG = 2r \sin\left(\frac{\theta}{2}\right) \]
where \( r \) is the radius and \( \theta \) is the central angle in degrees.
Here:
- \( r = 8 \) mm
- \( \theta = 60^\circ \)
First, calculate \( \frac{\theta}{2} \):
\[ \frac{\theta}{2} = \frac{60^\circ}{2} = 30^\circ \]
Now, use the sine function:
\[ \sin(30^\circ) = \frac{1}{2} \]
Now substitute \( r \) and \( \sin\left(30^\circ\right) \) into the chord length formula:
\[ FG = 2 \times 8 \times \sin(30^\circ) = 2 \times 8 \times \frac{1}{2} \]
\[ FG = 8 \text{ mm} \]
Thus, the exact measure of \( FG \) is:
\[ \boxed{8} \text{ mm} \]
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