To find the area of sector ABC given that \( m\angle ABC = 138^\circ \) and the radius \( AB = 14 \), we can use the formula for the area of a sector of a circle, which is given by:
\[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
Where:
- \( \theta \) is the angle in degrees,
- \( r \) is the radius.
Here, \( \theta = 138^\circ \) and \( r = 14 \).
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Calculate the area:
First, we will square the radius:
\[ r^2 = 14^2 = 196 \]
Now, substitute \( r^2 \) and \( \theta \) into the area formula:
\[ \text{Area of Sector} = \frac{138}{360} \times \pi \times 196 \]
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Calculate \( \frac{138}{360} \):
\[ \frac{138}{360} = \frac{23}{60} \approx 0.38333 \]
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Substituting into the area formula:
Now substituting this into the formula, we get:
\[ \text{Area of Sector} \approx 0.38333 \times \pi \times 196 \]
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Calculate the value:
Using \( \pi \approx 3.14159 \):
\[ \text{Area of Sector} \approx 0.38333 \times 3.14159 \times 196 \]
\[ \text{Area of Sector} \approx 0.38333 \times 615.75184 \approx 236.156 \]
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Rounding to the nearest hundredth:
The area of sector ABC, rounded to the nearest hundredth, is:
\[ \boxed{236.16} \]