To find the area of sector \( BCD \) in circle \( C \) where \( m\angle BCD = 74^\circ \) and the radius \( BC = 12 \), we can use the formula for the area of a sector, which is given by:
\[ \text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 \]
where \( \theta \) is the angle in degrees and \( r \) is the radius.
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Substitute the known values:
- \( \theta = 74^\circ \)
- \( r = 12 \)
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Calculate the area: \[ \text{Area} = \frac{74}{360} \times \pi \times (12)^2 \]
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Calculate \( (12)^2 \): \[ 12^2 = 144 \]
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Now substitute this into the area formula: \[ \text{Area} = \frac{74}{360} \times \pi \times 144 \]
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Calculate \( \frac{74}{360} \): \[ \frac{74}{360} \approx 0.20555556 \]
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Now calculate the area: \[ \text{Area} \approx 0.20555556 \times \pi \times 144 \] \[ \text{Area} \approx 0.20555556 \times 452.389342 \] (\( \pi \approx 3.14159265 \))
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Calculating \( 0.20555556 \times 452.389342 \) gives: \[ \text{Area} \approx 92.890657 \]
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Round to the nearest hundredth: \[ \text{Area} \approx 92.89 \]
Thus, the area of sector \( BCD \) is approximately \( \boxed{92.89} \).