1. **State the problem:** We are given a trapezoid QRTU with midsegment PS. The lengths are: - $TU = w$ - $PS = w + 10$ - $QR = -w + 50$ We need to find the value of $w$. 2. **Recall the midsegment formula for trapezoids:** The midsegment length is the average of the lengths of the two bases: $$PS = \frac{TU + QR}{2}$$ 3. **Substitute the given expressions:** $$w + 10 = \frac{w + (-w + 50)}{2}$$ 4. **Simplify inside the numerator:** $$w + 10 = \frac{\cancel{w} - \cancel{w} + 50}{2} = \frac{50}{2}$$ 5. **Simplify the fraction:** $$w + 10 = 25$$ 6. **Solve for $w$:** $$w = 25 - 10$$ $$w = 15$$ **Final answer:** $$\boxed{15}$$