Question

consider parallelogram wxyz below.
note that wxyz has vertices w(-2, -6), x(2, 4), y(-3, 6), and z(-7, -4).
answer the following to determine if the parallelogram is a rectangle, rhombus, square, or none of these.
(a) find the length of \\(\overline{wx}\\) and the length of a side adjacent to \\(\overline{wx}\\).
give exact answers (not decimal approximations).
length of \\(\overline{wx}\\):
length of side adjacent to \\(\overline{wx}\\):
(b) find the slope of \\(\overline{wx}\\) and the slope of a side adjacent to \\(\overline{wx}\\)
slope of \\(\overline{wx}\\):
slope of side adjacent to \\(\overline{wx}\\):
(c) from parts (a) and (b), what can be conclude about parallelogram wxyz? click all that apply
wxyz is a rectangle
wxyz is a rhombus
wxyz is a square
wxyz is none of these

Explanation:

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

Use distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
For $W(-2,-6)$ and $X(2,4)$:

$$\begin{align*} \overline{WX}&=\sqrt{(2-(-2))^2+(4-(-6))^2}\\ &=\sqrt{(4)^2+(10)^2}\\ &=\sqrt{16+100}\\ &=\sqrt{116}\\ &=2\sqrt{29} \end{align*}$$

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

Use distance formula for $X(2,4)$ and $Y(-3,6)$:

$$\begin{align*} \overline{XY}&=\sqrt{(-3-2)^2+(6-4)^2}\\ &=\sqrt{(-5)^2+(2)^2}\\ &=\sqrt{25+4}\\ &=\sqrt{29} \end{align*}$$

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

Use slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
For $W(-2,-6)$ and $X(2,4)$:

$$ m_{WX}=\frac{4-(-6)}{2-(-2)}=\frac{10}{4}=\frac{5}{2} $$

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

Use slope formula for $X(2,4)$ and $Y(-3,6)$:

$$ m_{XY}=\frac{6-4}{-3-2}=\frac{2}{-5}=-\frac{2}{5} $$

Step5: Classify the parallelogram

Check if slopes are negative reciprocals: $\frac{5}{2} \times -\frac{2}{5} = -1$, so sides are perpendicular. Check side lengths: $2\sqrt{29}
eq \sqrt{29}$, so sides are not equal. A parallelogram with perpendicular sides and unequal adjacent sides is a rectangle.

Answer:

(a)
Length of $\overline{WX}$: $2\sqrt{29}$
Length of side adjacent to $\overline{WX}$: $\sqrt{29}$

(b)
Slope of $\overline{WX}$: $\frac{5}{2}$
Slope of side adjacent to $\overline{WX}$: $-\frac{2}{5}$

(c)
WXYZ is a rectangle