1. **Problem Statement:** Find the length of segment $UV$ in the given circle diagram where angles and other segments are provided. 2. **Given:** The circle has points $T$, $U$, $V$, $C$ with angles $34^\circ$, $34^\circ$ and a segment length of 5. 3. **Formula and Rules:** In circle geometry, if two angles subtended by chords are equal, the chords are equal in length. Also, the Law of Cosines can be used if needed. 4. **Step-by-step solution:** - Since angles at $U$ and $V$ are both $34^\circ$, the chords $TU$ and $TV$ are equal. - Given $TC = 5$, and the two angles $34^\circ$ at $U$ and $V$, triangle $TUV$ is isosceles with $TU = TV$. - To find $UV$, use the Law of Cosines in triangle $TUV$: $$UV^2 = TU^2 + TV^2 - 2 \cdot TU \cdot TV \cdot \cos(\angle UTV)$$ - Since $TU = TV = 5$ and $\angle UTV = 180^\circ - 34^\circ - 34^\circ = 112^\circ$: $$UV^2 = 5^2 + 5^2 - 2 \cdot 5 \cdot 5 \cdot \cos(112^\circ)$$ $$UV^2 = 25 + 25 - 50 \cdot \cos(112^\circ)$$ - Calculate $\cos(112^\circ) \approx -0.3746$: $$UV^2 = 50 - 50 \cdot (-0.3746) = 50 + 18.73 = 68.73$$ - Therefore: $$UV = \sqrt{68.73} \approx 8.29$$ 5. **Answer:** $$\boxed{UV \approx 8.29}$$