To find the length of segment \( \overline{FG} \) on the circumference of the circle, we can use the properties of circles and trigonometry. Given that \( m\angle FCE = 60^\circ \) and the radius \( r = 8 , \text{mm} \), we can proceed as follows:
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Identify the central angle: The angle \( \angle FCE \) is the angle subtended at the center of the circle by the arc \( \overline{FE} \). Since \( m\angle FCE = 60^\circ \), the arc \( \overline{FE} \) also subtends a central angle of the same measure.
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Calculate the length of the arc \( \overline{FE} \): The length of an arc of a circle is given by the formula: \[ \text{Arc Length} = r \cdot \theta \] where \( \theta \) is in radians. We first need to convert \( 60^\circ \) to radians: \[ 60^\circ = \frac{60 \times \pi}{180} = \frac{\pi}{3} \text{ radians} \]
Now we can calculate the arc length: \[ \text{Arc Length} = 8 , \text{mm} \cdot \frac{\pi}{3} = \frac{8\pi}{3} , \text{mm} \]
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Length of \( \overline{FG} \): The points \( F \) and \( G \) subtend the same angle \( \angle FCE \). Since \( FG \) lies on the same arc as \( FE \), the arc length \( FG \) subtends the same angle. Therefore, we also have another arc \( \overline{FG} \) with the same measure for the other subtended \( G \).
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Compute \( \overline{FG} \): Since \( FG \) subtends the same angle and \( G \) is positioned diametrically opposite (if \( m\angle FCG = 60^\circ + 60^\circ \)), we can find segment \( FG \) using the properties of circles where \( FG \) resembles a triangle, considering horizontal areas.
To find the length of \( \overline{FG} \), we can also use the chord length formula: \[ \text{Chord Length} = 2r \sin\left(\frac{\theta}{2}\right) \] where \( r \) is the radius and \( \theta = 60^\circ \): \[ \text{Chord Length} = 2 \cdot 8 \sin\left(\frac{60^\circ}{2}\right) = 16 \sin(30^\circ) \] Since \( \sin(30^\circ) = \frac{1}{2} \), \[ \text{Chord Length} = 16 \cdot \frac{1}{2} = 8 , \text{mm} \]
Thus, the exact measure of \( \overline{FG} \) is: \[ \boxed{8} , \text{mm} \]
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