Question

consider parallelogram wxyz below.

note that wxyz has vertices $w(2, -7)$, $x(-6, -2)$, $y(-1, 6)$, and $z(7, 1)$.
answer the following to determine if the parallelogram is a rectangle, rhombus, square, or none of these.

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

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

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

Explanation:

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

Use distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
For $X(-6,-2)$ and $Y(-1,6)$:

$$\begin{align*} \overline{XY}&=\sqrt{(-1-(-6))^2+(6-(-2))^2}\\ &=\sqrt{(5)^2+(8)^2}\\ &=\sqrt{25+64}\\ &=\sqrt{89} \end{align*}$$

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

Adjacent side to $\overline{XY}$ is $\overline{YZ}$. Use distance formula for $Y(-1,6)$ and $Z(7,1)$:

$$\begin{align*} \overline{YZ}&=\sqrt{(7-(-1))^2+(1-6)^2}\\ &=\sqrt{(8)^2+(-5)^2}\\ &=\sqrt{64+25}\\ &=\sqrt{89} \end{align*}$$

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

Use slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$

$$\begin{align*} m_{\overline{XY}}&=\frac{6-(-2)}{-1-(-6)}\\ &=\frac{8}{5} \end{align*}$$

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

Use slope formula for $\overline{YZ}$:

$$\begin{align*} m_{\overline{YZ}}&=\frac{1-6}{7-(-1)}\\ &=\frac{-5}{8} \end{align*}$$

Step5: Analyze properties for classification

Check if sides are perpendicular: multiply slopes: $\frac{8}{5} \times \frac{-5}{8}=-1$, so angles are right angles. Sides are equal ($\sqrt{89}=\sqrt{89}$). A parallelogram with 4 right angles and 4 equal sides is a square (which is also a rectangle and rhombus).

Answer:

s:
(a)
Length of $\overline{XY}$: $\sqrt{89}$
Length of side adjacent to $\overline{XY}$: $\sqrt{89}$

(b)
Slope of $\overline{XY}$: $\frac{8}{5}$
Slope of side adjacent to $\overline{XY}$: $-\frac{5}{8}$

(c)

• WXYZ is a rectangle.
• WXYZ is a rhombus.
• WXYZ is a square.