Question

5
use the slope formula to find the slope of each side.
slope of \\(\overline{wx} = \frac{4 - 1}{3 - (-1)} = \\)
slope of \\(\overline{xy} = \frac{0 - 4}{6 - 3} = \\)
slope of \\(\overline{yz} = \frac{-3 - 0}{2 - 6} = \frac{3}{4}\\)
slope of \\(\overline{wz} = \frac{-3 - 1}{2 - (-1)} = \frac{-4}{3}\\)
therefore, in quadrilateral wxyz,
therefore, wxyz is a square.

Explanation:

Step1: Calculate slope of $\overline{WX}$

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For $\overline{WX}$, $y_2 = 4$, $y_1 = 1$, $x_2 = 3$, $x_1 = -1$. So $\frac{4 - 1}{3 - (-1)}=\frac{3}{4}$.

Step2: Calculate slope of $\overline{XY}$

For $\overline{XY}$, $y_2 = 0$, $y_1 = 4$, $x_2 = 6$, $x_1 = 3$. So $\frac{0 - 4}{6 - 3}=\frac{-4}{3}$.

Step3: Analyze slopes for quadrilateral

Slope of $\overline{WX}=\frac{3}{4}$, slope of $\overline{YZ}=\frac{3}{4}$ (so $\overline{WX}\parallel\overline{YZ}$). Slope of $\overline{XY}=-\frac{4}{3}$, slope of $\overline{WZ}=-\frac{4}{3}$ (so $\overline{XY}\parallel\overline{WZ}$). Also, product of slope of $\overline{WX}$ and $\overline{XY}$ is $\frac{3}{4}\times(-\frac{4}{3})=-1$, so they are perpendicular. Similarly for other adjacent sides. So opposite sides are parallel and adjacent sides are perpendicular (and we can check lengths too, but from slopes, opposite sides parallel, adjacent perpendicular, so it's a rectangle, and if sides equal, square. But from slope calculations, first two slopes: $\overline{WX}$ slope $\frac{3}{4}$, $\overline{XY}$ slope $-\frac{4}{3}$, $\overline{YZ}$ slope $\frac{3}{4}$, $\overline{WZ}$ slope $-\frac{4}{3}$. So opposite sides parallel, adjacent perpendicular.

Answer:

Slope of $\overline{WX}=\boldsymbol{\frac{3}{4}}$, Slope of $\overline{XY}=\boldsymbol{-\frac{4}{3}}$, and in quadrilateral $WXYZ$, opposite sides are parallel ( $\overline{WX}\parallel\overline{YZ}$, $\overline{XY}\parallel\overline{WZ}$ ) and adjacent sides are perpendicular.