To find the area of sector \( ABC \) in circle \( B \), we will use the formula for the area of a sector:
\[ \text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
where \( \theta \) is the angle in degrees, and \( r \) is the radius of the circle.
From the problem:
- The measure of angle \( ABC \) is \( \theta = 126^\circ \).
- The length \( AB = 17 \) serves as the radius \( r = 17 \) (since we can consider \( AB \) as the radius of circle \( B \)).
Now, substituting the values into the area formula:
\[ \text{Area of sector} = \frac{126}{360} \times \pi \times (17)^2 \]
Calculating \( (17)^2 \):
\[ (17)^2 = 289 \]
Now, substituting this back into the formula gives us:
\[ \text{Area of sector} = \frac{126}{360} \times \pi \times 289 \]
Calculating \( \frac{126}{360} \):
\[ \frac{126}{360} = \frac{7}{20} = 0.35 \]
Now, the area becomes:
\[ \text{Area of sector} = 0.35 \times \pi \times 289 \]
Using an approximate value for \( \pi \) (approximately \( 3.14 \)):
\[ \text{Area of sector} \approx 0.35 \times 3.14 \times 289 \]
Calculating \( 0.35 \times 3.14 \):
\[ 0.35 \times 3.14 \approx 1.099 \]
Now, multiply \( 1.099 \) by \( 289 \):
\[ \text{Area of sector} \approx 1.099 \times 289 \approx 317.611 \]
Finally, rounding to the nearest hundredth, the area of sector \( ABC \) is approximately:
\[ \boxed{317.61} \]