Question

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

Explanation:

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

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
For $G(4,-4)$ and $H(1,3)$:
$m_{\overline{GH}}=\frac{3-(-4)}{1-4}=\frac{7}{-3}=-\frac{7}{3}$

Step2: Calculate slope of adjacent $\overline{HG}$ (or $\overline{GK}$; using $\overline{HJ}$, adjacent to $\overline{GH}$: $H(1,3)$ and $J(-4,2)$)

$m_{\overline{HJ}}=\frac{2-3}{-4-1}=\frac{-1}{-5}=\frac{1}{5}$

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

Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$d_{\overline{GH}}=\sqrt{(1-4)^2+(3-(-4))^2}=\sqrt{(-3)^2+(7)^2}=\sqrt{9+49}=\sqrt{58}$

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

$d_{\overline{HJ}}=\sqrt{(-4-1)^2+(2-3)^2}=\sqrt{(-5)^2+(-1)^2}=\sqrt{25+1}=\sqrt{26}$

Step5: Classify the parallelogram

1. Check if it's a rectangle: Product of slopes of adjacent sides should be $-1$.

$-\frac{7}{3} \times \frac{1}{5}=-\frac{7}{15}
eq -1$, so not a rectangle (and thus not a square).

1. Check if it's a rhombus: All sides must be equal. $\sqrt{58}

eq \sqrt{26}$, so not a rhombus.

Answer:

(a)
Slope of $\overline{GH}$: $-\frac{7}{3}$
Slope of side adjacent to $\overline{GH}$: $\frac{1}{5}$

(b)
Length of $\overline{GH}$: $\sqrt{58}$
Length of side adjacent to $\overline{GH}$: $\sqrt{26}$

(c)
GHJK is none of these