In circle \( S \) with \( m \angle R S T=128^{\circ} \) and \( R S=11 \) units, find the length of \( R T \). Round to the nearest hundredth.

1. Recognize that in circle \( S \) with center \( S \), the segments \( RS \) and \( ST \) are radii. Therefore, \( RS = ST = 11 \) units. 2. In triangle \( RST \), \(\angle RST = 128^\circ\) and the two sides adjacent to the angle (the radii) have equal length. This means triangle \( RST \) is an isosceles triangle. 3. To find the chord \( RT \) (which is opposite the angle \( \angle RST \)), apply the Law of Cosines: \[ RT^2 = RS^2 + ST^2 - 2 \cdot RS \cdot ST \cdot \cos(128^\circ) \] Substitute the known values: \[ RT^2 = 11^2 + 11^2 - 2 \cdot 11 \cdot 11 \cdot \cos(128^\circ) \] \[ RT^2 = 121 + 121 - 242 \cdot \cos(128^\circ) \] 4. Calculate \(\cos(128^\circ)\). Since \(128^\circ\) is an obtuse angle, its cosine is negative. Using a calculator: \[ \cos(128^\circ) \approx -0.61566 \] 5. Substitute the cosine value into the equation: \[ RT^2 = 242 - 242 \cdot (-0.61566) \] \[ RT^2 = 242 + 149.11 \quad \text{(approximately)} \] \[ RT^2 \approx 391.11 \] 6. Take the square root of both sides to find \( RT \): \[ RT \approx \sqrt{391.11} \approx 19.78 \] 7. Rounded to the nearest hundredth, the length of \( RT \) is approximately: \[ 19.78 \text{ units} \]