Math / Geometry

QuestionConsider parallelogram GHJK below.
Note that $G H J K$ has vertices $G(-4,2), H(1,3), J(4,-4)$ , 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 $\overline{H J}$ and the slope of a side adjacent to $\overline{H J}$ .
Slope of $\overline{H J}$ : $\square$
Slope of side adjacent to $\overline{H J}$ : $\square$ (b) Find the length of $\overline{H J}$ and the length of a side adjacent to $\overline{H J}$ . Give exact answers (not decimal approximations).
Length of $\overline{H J}$ : $\square$
Length of side adjacent to $\overline{H J}$ : $\square$ (c) From parts (a) and (b), what can we conclude about parallelogram $G H J K$ ? Check all that apply.

Studdy Solution

STEP 1

1. The coordinates of the vertices of the parallelogram are given as $G(-4,2), H(1,3), J(4,-4), K(-1,-5)$ .
2. To determine if the parallelogram is a rectangle, rhombus, or square, we need to check the slopes and lengths of the sides.

STEP 2

1. Calculate the slope of $\overline{HJ}$ .
2. Calculate the slope of a side adjacent to $\overline{HJ}$ .
3. Calculate the length of $\overline{HJ}$ .
4. Calculate the length of a side adjacent to $\overline{HJ}$ .
5. Analyze the properties of the parallelogram based on slopes and lengths.

STEP 3

To find the slope of $\overline{HJ}$ , use the formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ :
$\text{Slope of } \overline{HJ} = \frac{y_2 - y_1}{x_2 - x_1}$
For $H(1,3)$ and $J(4,-4)$ :
$\text{Slope of } \overline{HJ} = \frac{-4 - 3}{4 - 1} = \frac{-7}{3}$

STEP 4

To find the slope of a side adjacent to $\overline{HJ}$ , consider $\overline{GH}$ with points $G(-4,2)$ and $H(1,3)$ :
$\text{Slope of } \overline{GH} = \frac{3 - 2}{1 - (-4)} = \frac{1}{5}$

STEP 5

To find the length of $\overline{HJ}$ , use the distance formula:
$\text{Length of } \overline{HJ} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
For $H(1,3)$ and $J(4,-4)$ :
$\text{Length of } \overline{HJ} = \sqrt{(4 - 1)^2 + (-4 - 3)^2} = \sqrt{3^2 + (-7)^2} = \sqrt{9 + 49} = \sqrt{58}$

STEP 6

To find the length of a side adjacent to $\overline{HJ}$ , consider $\overline{GH}$ :
$\text{Length of } \overline{GH} = \sqrt{(1 - (-4))^2 + (3 - 2)^2} = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26}$

STEP 7

Analyze the properties: - Since the slopes of $\overline{HJ}$ and $\overline{GH}$ are not negative reciprocals, $GHJK$ is not a rectangle. - The lengths of $\overline{HJ}$ and $\overline{GH}$ are not equal, so $GHJK$ is not a rhombus. - Since it is neither a rectangle nor a rhombus, it cannot be a square. - Therefore, $GHJK$ is none of these.
(a) Slope of $\overline{HJ}$ : $\frac{-7}{3}$
Slope of side adjacent to $\overline{HJ}$ : $\frac{1}{5}$
(b) Length of $\overline{HJ}$ : $\sqrt{58}$
Length of side adjacent to $\overline{HJ}$ : $\sqrt{26}$
(c) Conclusion: $GHJK$ is none of these.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms