1. **State the problem:** We have a circle with center $T$, and points $U$ and $V$ on the circle. The angle $\angle UTV$ is $50^\circ$, and the area of the shaded sector formed by radii $TU$ and $TV$ is $\frac{5}{4}\pi$. We need to find the length of the chord $UV$ expressed as a fraction times $\pi$. 2. **Relevant formulas:** - Area of a sector: $$\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2$$ where $\theta$ is the central angle in degrees and $r$ is the radius. - Length of chord: $$UV = 2r \sin\left(\frac{\theta}{2}\right)$$ 3. **Find the radius $r$ using the sector area:** Given: $$\frac{\theta}{360^\circ} \pi r^2 = \frac{5}{4} \pi$$ Substitute $\theta = 50^\circ$: $$\frac{50}{360} \pi r^2 = \frac{5}{4} \pi$$ Divide both sides by $\pi$: $$\frac{50}{360} r^2 = \frac{5}{4}$$ Simplify fraction $\frac{50}{360} = \frac{5}{36}$: $$\frac{5}{36} r^2 = \frac{5}{4}$$ Divide both sides by $5$: $$\frac{\cancel{5}}{36} r^2 = \frac{\cancel{5}}{4} \implies \frac{1}{36} r^2 = \frac{1}{4}$$ Multiply both sides by 36: $$r^2 = 36 \times \frac{1}{4} = 9$$ Take square root: $$r = 3$$ 4. **Find chord length $UV$:** $$UV = 2r \sin\left(\frac{\theta}{2}\right) = 2 \times 3 \times \sin\left(\frac{50^\circ}{2}\right) = 6 \sin(25^\circ)$$ 5. **Express answer as fraction times $\pi$:** Since the problem asks for the length of $UV$ expressed as a fraction times $\pi$, and $UV$ is $6 \sin(25^\circ)$, we write: $$UV = \frac{6 \sin(25^\circ)}{1} \pi \times \frac{1}{\pi} = 6 \sin(25^\circ)$$ But $\sin(25^\circ)$ is irrational, so the exact expression is: $$UV = 6 \sin(25^\circ)$$ If the problem expects a fraction times $\pi$, the length is not naturally expressed with $\pi$ since chord length is linear, not area. **Final answer:** $$UV = 6 \sin(25^\circ)$$