Question

consider parallelogram $jklm$ below.

note that $jklm$ has vertices $j(-2, 4)$, $k(-6, -1)$, $l(-1, 3)$, and $m(3, 8)$.
answer the following to determine if the parallelogram is a rectangle, rhombus, square, or none of these.

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

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

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

Explanation:

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

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
For $K(-6,-1), L(-1,3)$:
$m_{\overline{KL}}=\frac{3-(-1)}{-1-(-6)}=\frac{4}{5}$

Step2: Calculate slope of adjacent side $\overline{JK}$

Adjacent side to $\overline{KL}$ is $\overline{JK}$ (vertices $J(-2,4), K(-6,-1)$):
$m_{\overline{JK}}=\frac{4-(-1)}{-2-(-6)}=\frac{5}{4}$

Step3: Calculate length of $\overline{KL}$

Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$d_{\overline{KL}}=\sqrt{(-1-(-6))^2+(3-(-1))^2}=\sqrt{5^2+4^2}=\sqrt{25+16}=\sqrt{41}$

Step4: Calculate length of adjacent side $\overline{JK}$

$d_{\overline{JK}}=\sqrt{(-2-(-6))^2+(4-(-1))^2}=\sqrt{4^2+5^2}=\sqrt{16+25}=\sqrt{41}$

Step5: Analyze properties for part (c)

1. Check for rectangle: Perpendicular sides have slopes that are negative reciprocals. $\frac{4}{5} \times \frac{5}{4}=1

eq -1$, so sides are not perpendicular → not a rectangle (or square, since squares are rectangles).

1. Check for rhombus: All sides of parallelogram are equal (we found $d_{\overline{KL}}=d_{\overline{JK}}=\sqrt{41}$, and in parallelograms opposite sides are equal, so all 4 sides are equal).

Answer:

(a)
Slope of $\overline{KL}$: $\frac{4}{5}$
Slope of side adjacent to $\overline{KL}$: $\frac{5}{4}$

(b)
Length of $\overline{KL}$: $\sqrt{41}$
Length of side adjacent to $\overline{KL}$: $\sqrt{41}$

(c)
$\boldsymbol{JKLM}$ is a rhombus.