In circle \( G \) with \( m \angle F G H=34^{\circ} \) and \( F G=10 \), find the area of sector FGH. Round to the nearest hundredth.

To find the area of sector FGH in circle \( G \), we can use the formula for the area of a sector, which is given by: \[ \text{Area} = \frac{\theta}{360} \times \pi r^2 \] where \( \theta \) is the angle in degrees and \( r \) is the radius of the circle. In this case, \( \theta = 34^\circ \) and \( r = FG = 10 \). First, we calculate the area using the formula: \[ \text{Area} = \frac{34}{360} \times \pi \times (10)^2 \] Calculating \( 10^2 \): \[ 10^2 = 100 \] Now substitute back into the formula: \[ \text{Area} = \frac{34}{360} \times \pi \times 100 \] Simplifying \( \frac{34}{360} \): \[ \frac{34}{360} = \frac{17}{180} \] So we can write: \[ \text{Area} = \frac{17}{180} \times \pi \times 100 = \frac{1700 \pi}{180} \] Now simplifying \( \frac{1700}{180} \): \[ \frac{1700}{180} = \frac{85}{9} \] Now substituting for \( \pi \approx 3.14159 \): \[ \text{Area} \approx \frac{85}{9} \times 3.14159 \approx 29.577 \text{ (approximately)} \] Rounding to the nearest hundredth: \[ \text{Area} \approx 29.58 \] Thus, the area of sector FGH is approximately \( \boxed{29.58} \).