consider parallelogram j k l m below.
note that j k l m has vertices j(-5, -5), k(1, 4), l(-2, 6), and m(-8, -3).
answer the following to determine if the parallelogram is a rectangle, rhombus, square, or none of these.
(a) find the length of $ \overline { j k } $ and the length of a side adjacent to $ \overline { j k } $.
give exact answers (not decimal approximations).
length of $ \overline { j k } $:
length of side adjacent to $ \overline { j k } $:
(b) find the slope of $ \overline { j k } $ and the slope of a side adjacent to $ \overline { j k } $.
slope of $ \overline { j k } $:
slope of side adjacent to $ \overline { j k } $:
(c) from parts (a) and (b), what can we conclude about parallelogram j k l m? check all that apply.
$ \square j k l m $ is a rectangle.
$ \square j k l m $ is a rhombus.
$ \square j k l m $ is a square.
$ \square j k l m $ is none of these.

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Accepted Answer

# Explanation:
## Step1: Calculate the length of $ \overline{JK} $
Use the distance formula $ d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} $. For $ J(-5,-5) $ and $ K(1,4) $, $ x_1=-5,y_1 = -5,x_2=1,y_2 = 4 $.
$$
\begin{align*}
d_{JK}&=\sqrt{(1-(-5))^2+(4 - (-5))^2}\
&=\sqrt{(1 + 5)^2+(4+5)^2}\
&=\sqrt{6^2+9^2}\
&=\sqrt{36 + 81}\
&=\sqrt{117}\
&=3\sqrt{13}
\end{align*}
$$
## Step2: Calculate the length of a side adjacent to $ \overline{JK} $ (e.g., $ \overline{KL} $)
For $ K(1,4) $ and $ L(-2,6) $, $ x_1 = 1,y_1=4,x_2=-2,y_2 = 6 $
$$
\begin{align*}
d_{KL}&=\sqrt{(-2 - 1)^2+(6 - 4)^2}\
&=\sqrt{(-3)^2+2^2}\
&=\sqrt{9+4}\
&=\sqrt{13}
\end{align*}
$$
## Step3: Calculate the slope of $ \overline{JK} $
Use the slope formula $ m=\frac{y_2-y_1}{x_2-x_1} $. For $ J(-5,-5) $ and $ K(1,4) $, $ m_{JK}=\frac{4-(-5)}{1-(-5)}=\frac{4 + 5}{1+5}=\frac{9}{6}=\frac{3}{2} $
## Step4: Calculate the slope of a side adjacent to $ \overline{JK} $ (e.g., $ \overline{KL} $)
For $ K(1,4) $ and $ L(-2,6) $, $ m_{KL}=\frac{6 - 4}{-2-1}=\frac{2}{-3}=-\frac{2}{3} $

# Answer:
(a) Length of $ \overline{JK}:3\sqrt{13} $, Length of side adjacent to $ \overline{JK}:\sqrt{13} $
(b) Slope of $ \overline{JK}:\frac{3}{2} $, Slope of side adjacent to $ \overline{JK}:-\frac{2}{3} $
(c) Since $ 3\sqrt{13}
eq\sqrt{13} $ (not a rhombus or square), and $ \frac{3}{2}\times(-\frac{2}{3})=- 1 $ (adjacent sides are perpendicular). So $ J K L M $ is a rectangle.
C. $ J K L M $ is a rectangle.

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Question

Explanation:

Step1: Calculate the length of \( \overline{JK} \)

Step2: Calculate the length of a side adjacent to \( \overline{JK} \) (e.g., \( \overline{KL} \))

Step3: Calculate the slope of \( \overline{JK} \)

Step4: Calculate the slope of a side adjacent to \( \overline{JK} \) (e.g., \( \overline{KL} \))

Answer: