deltamath.com DeltaMath Student Application SNAKE - Google Search Question Watch Video Show Examples In circle B with \( m \angle A B C=112^{\circ} \) and \( A B=17 \), find the area of sector ABC . Round to the nearest hundredth.

Let the circle have center \( B \). Since \( A \) lies on the circle, the distance \( AB \) is the radius of the circle, so \[ r = 17. \] The central angle \( \angle ABC \) is given as \[ 112^\circ. \] The area \( A_{\text{sector}} \) of a sector with central angle \( \theta \) (in degrees) and radius \( r \) is given by \[ A_{\text{sector}} = \frac{\theta}{360} \pi r^2. \] Substitute the given values: \[ A_{\text{sector}} = \frac{112}{360} \pi (17)^2. \] Calculate \( 17^2 \): \[ 17^2 = 289. \] Thus, \[ A_{\text{sector}} = \frac{112}{360} \pi \cdot 289. \] Simplify the fraction \(\frac{112}{360}\) by dividing the numerator and the denominator by \(8\): \[ \frac{112}{360} = \frac{14}{45}. \] Now the area becomes: \[ A_{\text{sector}} = \frac{14 \cdot 289 \pi}{45}. \] Multiply \(14 \times 289\): \[ 14 \times 289 = 4046. \] So, \[ A_{\text{sector}} = \frac{4046\pi}{45}. \] Now, compute the decimal approximation: \[ \frac{4046}{45} \approx 89.91, \] so \[ A_{\text{sector}} \approx 89.91 \pi. \] Using \(\pi \approx 3.14159\): \[ A_{\text{sector}} \approx 89.91 \times 3.14159 \approx 282.74. \] Thus, the area of sector \(ABC\) is approximately \[ \boxed{282.74}. \]