1. **State the problem:** We are given a trapezoid RTUW with midsegment $ \overline{SV} $. The lengths are given as $ RW = -2p + 65 $, $ SV = p + 50 $, and $ TU = p + 71 $. We need to find the value of $ p $. 2. **Recall the midsegment formula for trapezoids:** The midsegment length is the average of the lengths of the two bases: $$ SV = \frac{RW + TU}{2} $$ 3. **Set up the equation using the given expressions:** $$ p + 50 = \frac{(-2p + 65) + (p + 71)}{2} $$ 4. **Simplify the right side:** $$ p + 50 = \frac{-2p + 65 + p + 71}{2} = \frac{-p + 136}{2} $$ 5. **Multiply both sides by 2 to clear the denominator:** $$ 2(p + 50) = -p + 136 $$ Intermediate step showing cancellation: $$ \cancel{2}(p + 50) = -p + 136 $$ 6. **Distribute the 2 on the left:** $$ 2p + 100 = -p + 136 $$ 7. **Add $ p $ to both sides to collect $ p $ terms on one side:** $$ 2p + p + 100 = 136 $$ $$ 3p + 100 = 136 $$ 8. **Subtract 100 from both sides:** $$ 3p = 136 - 100 $$ $$ 3p = 36 $$ 9. **Divide both sides by 3 to solve for $ p $:** $$ p = \frac{36}{3} $$ Intermediate step showing cancellation: $$ p = \frac{\cancel{36}}{\cancel{3}} $$ 10. **Simplify:** $$ p = 12 $$ **Final answer:** $ p = 12 $