In circle \( S \) with \( m \angle R S T=66^{\circ} \) and \( R S=7 \) units, find the length of \( R T \). Round to the nearest hundredth.

To find the length of \( R T \) in circle \( S \) given that \( m \angle RST = 66^\circ \) and \( RS = 7 \) units, we can use the Law of Sines. In triangle \( RST \), we have: \[ \frac{RT}{\sin \angle RST} = \frac{RS}{\sin \angle RST'} \] Where \( \angle RST' \) is the angle opposite side \( RT \). Using the triangle property that the angles must sum to \( 180^\circ \), we can establish that the external angle at \( S \) (which is equal to \( \angle RST'\)) would depend on the other angles of the triangle which we ideally would need more information to solve completely. However, in our case, we can apply some trigonometric relationships based on the subtended angles if we assume that \( S \) is the center. Assuming we are looking for the chord's length, not just the triangle formed, we can use the formula for the chord length \( c \): \[ c = 2 \cdot r \cdot \sin\left( \frac{\theta}{2} \right) \] where \( r = RS = 7 \) units, and \( \theta = \angle RST = 66^\circ \). Calculating \( RT \): \[ RT = 2 \cdot 7 \cdot \sin\left( \frac{66^\circ}{2} \right) = 14 \cdot \sin(33^\circ) \] Using a calculator for \( \sin(33^\circ) \): \[ \sin(33^\circ) \approx 0.5446 \] Thus: \[ RT \approx 14 \cdot 0.5446 \approx 7.6244 \] Rounding to the nearest hundredth, we get: \[ RT \approx 7.62 \] So, the length of \( RT \) is approximately 7.62 units.