Question

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

Explanation:

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

Slope formula: $m = \frac{y_2-y_1}{x_2-x_1}$. For $L(4,4)$ and $M(-3,2)$:
$m_{\overline{LM}} = \frac{2-4}{-3-4} = \frac{-2}{-7} = \frac{2}{7}$

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

Adjacent side to $\overline{LM}$ is $\overline{MJ}$ (or $\overline{LK}$; using $M(-3,2)$ and $J(-5,-3)$):
$m_{\overline{MJ}} = \frac{-3-2}{-5-(-3)} = \frac{-5}{-2} = \frac{5}{2}$

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

Distance formula: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. For $L(4,4)$ and $M(-3,2)$:
$d_{\overline{LM}} = \sqrt{(-3-4)^2+(2-4)^2} = \sqrt{(-7)^2+(-2)^2} = \sqrt{49+4} = \sqrt{53} \approx 7.28$

Step4: Calculate length of adjacent $\overline{MJ}$

For $M(-3,2)$ and $J(-5,-3)$:
$d_{\overline{MJ}} = \sqrt{(-5-(-3))^2+(-3-2)^2} = \sqrt{(-2)^2+(-5)^2} = \sqrt{4+25} = \sqrt{29} \approx 5.39$

Step5: Classify the parallelogram

1. Check for rectangle: Product of slopes of adjacent sides $\frac{2}{7} \times \frac{5}{2} = \frac{5}{7}

eq -1$, so sides are not perpendicular (not a rectangle/square).

1. Check for rhombus: $\sqrt{53}

eq \sqrt{29}$, so sides are not congruent (not a rhombus/square).

Answer:

(A)
Slope of $\overline{LM}$: $\frac{2}{7}$
Slope of side adjacent to $\overline{LM}$: $\frac{5}{2}$

(B)
Length of $\overline{LM}$: $\sqrt{53}$ (or $\approx 7.28$)
Length of side adjacent to $\overline{LM}$: $\sqrt{29}$ (or $\approx 5.39$)

(C)
JKLM is none of these.