1. **State the problem:** Find the exact length of the midsegment of trapezoid $STUV$ with vertices $S(-2,4)$, $T(-2,-4)$, $U(3,-2)$, and $V(13,10)$. 2. **Recall the midsegment formula:** The midsegment (or median) of a trapezoid is the segment connecting the midpoints of the non-parallel sides. Its length is the average of the lengths of the two parallel sides. 3. **Identify the parallel sides:** Calculate slopes of sides to find which are parallel. - Slope $ST = \frac{-4-4}{-2-(-2)} = \frac{-8}{0}$ undefined (vertical line). - Slope $UV = \frac{10-(-2)}{13-3} = \frac{12}{10} = 1.2$. - Slope $TU = \frac{-2-(-4)}{3-(-2)} = \frac{2}{5} = 0.4$. - Slope $SV = \frac{10-4}{13-(-2)} = \frac{6}{15} = 0.4$. Since $TU$ and $SV$ have the same slope $0.4$, these are the parallel sides. 4. **Calculate lengths of parallel sides $TU$ and $SV$: - Length $TU = \sqrt{(3-(-2))^2 + (-2-(-4))^2} = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29}$. - Length $SV = \sqrt{(13-(-2))^2 + (10-4)^2} = \sqrt{15^2 + 6^2} = \sqrt{225 + 36} = \sqrt{261}$. 5. **Calculate the midsegment length:** $$ \text{Midsegment length} = \frac{TU + SV}{2} = \frac{\sqrt{29} + \sqrt{261}}{2} $$ 6. **Final answer:** The exact length of the midsegment is $$ \boxed{\frac{\sqrt{29} + \sqrt{261}}{2}} $$