1. **Problem statement:** Given that QS is a midsegment of triangle PRT, and the lengths PT = $x + 54$ and QS = $x + 25$, find the value of $x$. 2. **Recall the midsegment theorem:** A midsegment in a triangle is parallel to one side and its length is half the length of that side. 3. **Apply the theorem:** Since QS is a midsegment parallel to PT, we have: $$QS = \frac{1}{2} PT$$ 4. **Substitute the given expressions:** $$x + 25 = \frac{1}{2} (x + 54)$$ 5. **Solve for $x$:** Multiply both sides by 2: $$2(x + 25) = x + 54$$ $$2x + 50 = x + 54$$ Subtract $x$ from both sides: $$2x - x + 50 = 54$$ $$x + 50 = 54$$ Subtract 50 from both sides: $$x = 54 - 50$$ $$x = 4$$ 6. **Answer:** The value of $x$ is 4.