To find the length of \( \overline{FG} \) (the chord between points \( F \) and \( G \)), we can use the central angle \( \angle FCE = 60^\circ \).
Since \( F \) and \( G \) both lie on the circle centered at \( C \), we can apply the following formula for the length of the chord \( FG \):
\[ FG = 2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right) \]
where:
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle (in this case \( \angle FCE \)).
Given:
- \( r = 8 , \text{mm} \)
- \( \theta = 60^\circ \)
First, we calculate \( \frac{\theta}{2} \):
\[ \frac{60^\circ}{2} = 30^\circ \]
Next, we substitute the values into the chord length formula:
\[ FG = 2 \cdot 8 \cdot \sin(30^\circ) \]
We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]
Now substitute this value in:
\[ FG = 2 \cdot 8 \cdot \frac{1}{2} \] \[ FG = 2 \cdot 8 \cdot 0.5 \] \[ FG = 8 , \text{mm} \]
Thus, the exact measure of \( \overline{FG} \) is:
\[ \boxed{8 , \text{mm}} \]
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