Question

consider parallelogram wxyz below.

note that wxyz has vertices $w(-2, 6)$, $x(-8, -3)$, $y(-5, -5)$, and $z(1, 4)$.
answer the following to determine if the parallelogram is a rectangle, rhombus, square, or none of these.

(a) find the length of $overline{yz}$ and the length of a side adjacent to $overline{yz}$.
give exact answers (not decimal approximations).
length of $overline{yz}$:
length of side adjacent to $overline{yz}$:

(b) find the slope of $overline{yz}$ and the slope of a side adjacent to $overline{yz}$
slope of $overline{yz}$:
slope of side adjacent to $overline{yz}$:

(c) from parts (a) and (b), what can we conclude about parallelogram wxyz? check all that apply.
$square$ wxyz is a rectangle.
$square$ wxyz is a rhombus.
$square$ wxyz is a square
$square$ wxyz is none of these.

Explanation:

Step1: Calculate length of $\overline{YZ}$

Use distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
For $Y(-5,-5)$ and $Z(1,4)$:
$$\overline{YZ}=\sqrt{(1-(-5))^2+(4-(-5))^2}=\sqrt{6^2+9^2}=\sqrt{36+81}=\sqrt{117}=3\sqrt{13}$$

Step2: Calculate length of adjacent side $\overline{XY}$

Use distance formula for $X(-8,-3)$ and $Y(-5,-5)$:
$$\overline{XY}=\sqrt{(-5-(-8))^2+(-5-(-3))^2}=\sqrt{3^2+(-2)^2}=\sqrt{9+4}=\sqrt{13}$$

Step3: Calculate slope of $\overline{YZ}$

Use slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
For $Y(-5,-5)$ and $Z(1,4)$:
$$m_{YZ}=\frac{4-(-5)}{1-(-5)}=\frac{9}{6}=\frac{3}{2}$$

Step4: Calculate slope of adjacent $\overline{XY}$

Use slope formula for $X(-8,-3)$ and $Y(-5,-5)$:
$$m_{XY}=\frac{-5-(-3)}{-5-(-8)}=\frac{-2}{3}=-\frac{2}{3}$$

Step5: Classify the parallelogram

Check side lengths: $\overline{YZ}
eq \overline{XY}$, so not a rhombus/square.
Check slopes: $m_{YZ} \times m_{XY} = \frac{3}{2} \times -\frac{2}{3} = -1$, so sides are perpendicular.

Answer:

(a)
Length of $\overline{YZ}$: $3\sqrt{13}$
Length of side adjacent to $\overline{YZ}$: $\sqrt{13}$

(b)
Slope of $\overline{YZ}$: $\frac{3}{2}$
Slope of side adjacent to $\overline{YZ}$: $-\frac{2}{3}$

(c)
WXYZ is a rectangle.