1. **State the problem:** We are given triangle $QSU$ with $RT$ as the midsegment connecting the midpoints of sides $QU$ and $US$. We know $QU = w + 49$ and $RT = w + 17$. We need to find the value of $w$. 2. **Recall the midsegment theorem:** The midsegment in a triangle is parallel to the third side and its length is half the length of that side. Here, $RT$ is the midsegment corresponding to side $QU$, so: $$RT = \frac{1}{2} QU$$ 3. **Set up the equation using the given expressions:** $$w + 17 = \frac{1}{2} (w + 49)$$ 4. **Solve for $w$:** Multiply both sides by 2 to clear the fraction: $$2(w + 17) = w + 49$$ Simplify the left side: $$2w + 34 = w + 49$$ Subtract $w$ from both sides: $$2w - w + 34 = 49$$ $$w + 34 = 49$$ Subtract 34 from both sides: $$w = 49 - 34$$ $$w = 15$$ 5. **Final answer:** $$\boxed{15}$$