1. The problem involves two circle diagrams with given angles and lengths. 2. We are given that $UV = 8.29$ and need to find $QR$. 3. Both circles have an angle of $120^\circ$ and radii given or implied. 4. In the left circle, $C$ is the center, $CS = 7$ is the radius, and $\angle QCR = 120^\circ$. 5. The chord length $QR$ in a circle can be found using the formula for chord length: $$QR = 2r \sin\left(\frac{\theta}{2}\right)$$ where $r$ is the radius and $\theta$ is the central angle in radians or degrees. 6. Substitute $r = 7$ and $\theta = 120^\circ$: $$QR = 2 \times 7 \times \sin\left(\frac{120^\circ}{2}\right) = 14 \times \sin(60^\circ)$$ 7. We know $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, so: $$QR = 14 \times \frac{\sqrt{3}}{2} = 7\sqrt{3}$$ 8. Numerically, $\sqrt{3} \approx 1.732$, so: $$QR \approx 7 \times 1.732 = 12.124$$ **Final answer:** $$QR = 7\sqrt{3} \approx 12.12$$