Question

in circle y, what is $mwidehat{tu}$?
○ $82^circ$
○ $100^circ$
○ $106^circ$
○ $118^circ$
the circle y has arcs: $mwidehat{rs}=106^circ$, and the angle between chords $st$ and $sr$ is $94^circ$.

Explanation:

Step1: Find arc $RT$ measure

The measure of a tangent-chord angle is half its intercepted arc.
$m\angle STR = \frac{1}{2}m\overset{\frown}{RT}$
$94^\circ = \frac{1}{2}m\overset{\frown}{RT}$
$m\overset{\frown}{RT} = 94^\circ \times 2 = 188^\circ$

Step2: Find arc $TU$ measure

Total circle degrees = $360^\circ$.
$m\overset{\frown}{RT} + m\overset{\frown}{RS} + m\overset{\frown}{TU} = 360^\circ$
$188^\circ + 106^\circ + m\overset{\frown}{TU} = 360^\circ$
$m\overset{\frown}{TU} = 360^\circ - 188^\circ - 106^\circ$

Answer:

$66^\circ$
Note: This result does not match the provided options, indicating a potential error in the problem's given values or options. The calculation follows the tangent-chord angle theorem and total circle angle sum rule.