1. **State the problem:** Show that quadrilateral WXYZ with vertices W(1,4), X(1,8), Y(-3,9), Z(-3,3) is a trapezoid and decide if it is isosceles. 2. **Recall the trapezoid definition:** A trapezoid has exactly one pair of parallel sides. 3. **Find slopes of all sides:** - Slope WX = \frac{8-4}{1-1} = \frac{4}{0} = \text{undefined (vertical line)} - Slope XY = \frac{9-8}{-3-1} = \frac{1}{-4} = -\frac{1}{4} - Slope YZ = \frac{3-9}{-3-(-3)} = \frac{-6}{0} = \text{undefined (vertical line)} - Slope ZW = \frac{4-3}{1-(-3)} = \frac{1}{4} 4. **Identify parallel sides:** WX and YZ both have undefined slope (vertical), so WX \parallel YZ. 5. **Check if trapezoid is isosceles:** Non-parallel sides are XY and ZW. - Slope XY = -\frac{1}{4} - Slope ZW = \frac{1}{4} 6. **Calculate lengths of legs XY and ZW:** - Length XY = \sqrt{(-3-1)^2 + (9-8)^2} = \sqrt{(-4)^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} - Length ZW = \sqrt{(1-(-3))^2 + (4-3)^2} = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} 7. Since legs XY and ZW are equal in length, trapezoid WXYZ is isosceles. 8. **Final conclusion:** Quadrilateral WXYZ is a trapezoid with one pair of parallel sides WX and YZ, and it is isosceles because its legs XY and ZW are congruent. --- 9. **State the problem:** Find the length of the midsegment of trapezoid ABCD with vertices A(2,0), B(8,-4), C(12,2), D(0,10). 10. **Recall midsegment formula:** Midsegment length = \frac{1}{2} (base_1 + base_2) 11. **Find slopes to identify bases:** - AB slope = \frac{-4-0}{8-2} = \frac{-4}{6} = -\frac{2}{3} - BC slope = \frac{2-(-4)}{12-8} = \frac{6}{4} = \frac{3}{2} - CD slope = \frac{10-2}{0-12} = \frac{8}{-12} = -\frac{2}{3} - DA slope = \frac{0-10}{2-0} = \frac{-10}{2} = -5 12. Bases are AB and CD since they have equal slopes (-\frac{2}{3}). 13. Calculate lengths: - AB = \sqrt{(8-2)^2 + (-4-0)^2} = \sqrt{6^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} - CD = \sqrt{(0-12)^2 + (10-2)^2} = \sqrt{(-12)^2 + 8^2} = \sqrt{144 + 64} = \sqrt{208} = 4\sqrt{13} 14. Midsegment length = \frac{1}{2} (AB + CD) = \frac{1}{2} (2\sqrt{13} + 4\sqrt{13}) = \frac{1}{2} (6\sqrt{13}) = 3\sqrt{13} 15. **Final answer:** The midsegment length is $3\sqrt{13}$. --- 16. **State the problem:** Find the length of the midsegment of trapezoid STUV with vertices S(-2,4), T(-2,-4), U(3,-2), V(13,10). 17. **Identify bases by slope:** - ST slope = \frac{-4-4}{-2-(-2)} = \frac{-8}{0} = \text{undefined} - TU slope = \frac{-2-(-4)}{3-(-2)} = \frac{2}{5} - UV slope = \frac{10-(-2)}{13-3} = \frac{12}{10} = \frac{6}{5} - VS slope = \frac{4-10}{-2-13} = \frac{-6}{-15} = \frac{2}{5} 18. Bases are TU and VS since both have slope \frac{2}{5}. 19. Calculate lengths: - TU = \sqrt{(3-(-2))^2 + (-2-(-4))^2} = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} - VS = \sqrt{(-2-13)^2 + (4-10)^2} = \sqrt{(-15)^2 + (-6)^2} = \sqrt{225 + 36} = \sqrt{261} 20. Midsegment length = \frac{1}{2} (TU + VS) = \frac{1}{2} (\sqrt{29} + \sqrt{261}) 21. **Final answer:** The midsegment length is $\frac{\sqrt{29} + \sqrt{261}}{2}$.