1. **State the problem:** We are given a circle with center Z and points Q, R, S, U, V on or inside the circle. We know QR = 4x - 2, SR = 10, ZU = 8, and ZV = 8. We need to find the values of $x$ and $VS$. 2. **Important properties and formulas:** - Since Z is the center, ZU and ZV are radii of the circle, so $ZU = ZV = 8$. - The perpendicular from the center to a chord bisects the chord. - If V and U are the feet of perpendiculars from Z to chords, then V and U are midpoints of those chords. 3. **Find $x$:** - Since V is the midpoint of chord SR, $SV = VR$. - Given $SR = 10$, so $SV = VR = \frac{10}{2} = 5$. - Given $QR = 4x - 2$, and since Q, R, S are points on the circle, and V is midpoint of SR, we consider the relationship between QR and SR. - However, the problem does not provide direct relation between QR and SR, so we focus on the chord SR and the radius. 4. **Find $VS$:** - Since V is midpoint of SR, $VS = 5$. 5. **Summary:** - $x$ is not directly solvable with given data unless more information is provided. - $VS = 5$ by midpoint property. **Final answers:** $$x = \text{not enough information}$$ $$VS = 5$$