We are given the coordinates of the vertices of a quadrilateral UVWX: $U(6,-2)$, $V(1,3)$, $W(-7,6)$, and $X(-2,1)$. We need to find the slopes and lengths of the sides UV, VW, WX, and XU and then determine the type of quadrilateral.

1. Problem Description

We are given the coordinates of the vertices of a quadrilateral UVWX: $U(6,-2)$, $V(1,3)$, $W(-7,6)$, and $X(-2,1)$. We need to find the slopes and lengths of the sides UV, VW, WX, and XU and then determine the type of quadrilateral.

2. Solution Steps

First, we find the slope of each side. The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Slope of UV:

$m_{UV} = \frac{3 - (-2)}{1 - 6} = \frac{5}{-5} = -1$

Slope of VW:

$m_{VW} = \frac{6 - 3}{-7 - 1} = \frac{3}{-8} = -\frac{3}{8}$

Slope of WX:

$m_{WX} = \frac{1 - 6}{-2 - (-7)} = \frac{-5}{5} = -1$

Slope of XU:

$m_{XU} = \frac{-2 - 1}{6 - (-2)} = \frac{-3}{8} = -\frac{3}{8}$

Next, we find the length of each side. The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Length of UV:

$d_{UV} = \sqrt{(1 - 6)^2 + (3 - (-2))^2} = \sqrt{(-5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}$

Length of VW:

$d_{VW} = \sqrt{(-7 - 1)^2 + (6 - 3)^2} = \sqrt{(-8)^2 + (3)^2} = \sqrt{64 + 9} = \sqrt{73}$

Length of WX:

$d_{WX} = \sqrt{(-2 - (-7))^2 + (1 - 6)^2} = \sqrt{(5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}$

Length of XU:

$d_{XU} = \sqrt{(6 - (-2))^2 + (-2 - 1)^2} = \sqrt{(8)^2 + (-3)^2} = \sqrt{64 + 9} = \sqrt{73}$

Since UV = WX and VW = XU, we have a parallelogram.

Now, we check if it is a rectangle. For a rectangle, adjacent sides must be perpendicular.

$m_{UV} \times m_{VW} = (-1) \times (-\frac{3}{8}) = \frac{3}{8} \ne -1$

Thus, the quadrilateral is not a rectangle.

It's a parallelogram.

3. Final Answer

slope of UV = -1

slope of VW = -3/8

slope of WX = -1

slope of XU = -3/8

length of UV = $5\sqrt{2}$

length of VW = $\sqrt{73}$

length of WX = $5\sqrt{2}$

length of XU = $\sqrt{73}$

Quadrilateral UVWX can BEST be described as parallelogram