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 \( RT \) in circle \( S \) with \( m \angle RST = 66^\circ \) and \( RS = 7 \) units, we can use the Law of Cosines. First, we recognize that the angle \( RST \) is an inscribed angle that subtends arc \( RT \). The length \( RT \) can be conceptualized as the third side in triangle \( RST \). According to the Law of Cosines: \[ RT^2 = RS^2 + ST^2 - 2 \times RS \times ST \times \cos(m \angle RST) \] With \( RS = 7 \) and \( ST = 7 \) (since both are radii of circle \( S \)), the lengths \( RS \) and \( ST \) are equal. Substituting the known values: \[ RT^2 = 7^2 + 7^2 - 2 \times 7 \times 7 \times \cos(66^\circ) \] Now, calculate \( 7^2 \): \[ RT^2 = 49 + 49 - 2 \times 49 \times \cos(66^\circ) \] This simplifies to: \[ RT^2 = 98 - 98 \times \cos(66^\circ) \] Using the cosine of \( 66^\circ \): \(\cos(66^\circ) \approx 0.4067\) Now substitute the cosine value: \[ RT^2 = 98 - 98 \times 0.4067 \] Calculating further: \[ RT^2 = 98 - 39.8006 \approx 58.1994 \] Taking the square root to find \( RT \): \[ RT \approx \sqrt{58.1994} \approx 7.62 \] Thus, the length of \( RT \) rounded to the nearest hundredth is approximately \( 7.62 \) units.