Trapezoid Wxyz A64A87

1. **State the problem:** Show that the quadrilateral with vertices $W(1,4)$, $X(1,8)$, $Y(-3,9)$, and $Z(-3,3)$ is a trapezoid and determine if it is isosceles. 2. **Recall the trapezoid definition:** A trapezoid is a quadrilateral with at least one pair of parallel sides. 3. **Find slopes of all sides:** - Slope of $WX = \frac{8-4}{1-1} = \frac{4}{0}$ which is undefined (vertical line). - Slope of $XY = \frac{9-8}{-3-1} = \frac{1}{-4} = -\frac{1}{4}$. - Slope of $YZ = \frac{3-9}{-3-(-3)} = \frac{-6}{0}$ which is undefined (vertical line). - Slope of $ZW = \frac{4-3}{1-(-3)} = \frac{1}{4}$. 4. **Identify parallel sides:** - $WX$ and $YZ$ both have undefined slopes (vertical), so $WX \parallel YZ$. 5. **Check if trapezoid is isosceles:** - Calculate lengths of legs $XY$ and $ZW$: $$XY = \sqrt{(-3-1)^2 + (9-8)^2} = \sqrt{(-4)^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17}$$ $$ZW = \sqrt{(1+3)^2 + (4-3)^2} = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17}$$ - Since $XY = ZW$, the trapezoid is isosceles. **Final answer:** The quadrilateral $WXYZ$ is a trapezoid with $WX \parallel YZ$ and it is isosceles because the non-parallel sides are equal in length.