Math / Geometry

Question,nsider parallelogram $J K L M$ below.
Note that $J K L M$ has vertices $J(-6,-2), K(-1,6), L(7,1)$ , and Answer the following to determine if the parallelogram is a rectangle, (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 $\overline{J K}$ : $\square$
Length of side adjacent to $\overline{J K}$ : $\square$

Studdy Solution

STEP 1

What is this asking? We need to find the lengths of two sides of a parallelogram on a graph, specifically side JK and a side next to it (either JM or KL). Watch out! Don't mix up the x and y coordinates in the distance formula!
Also, remember we want exact answers, so keep those square roots!

STEP 2

1. Find the length of JK.
2. Find the length of KL.

STEP 3

Alright, let's kick things off with the distance formula!
This formula helps us find the distance between two points on a graph.
Remember, it's like the Pythagorean theorem in disguise!
The distance formula is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

STEP 4

Our dynamic duo of points is J at $(-6, -2)$ and K at $(-1, 6)$ .
Let's plug these coordinates into our super-powered distance formula!
We'll call $(-6, -2)$ point 1, so $x_1 = -6$ and $y_1 = -2$ .
And $(-1, 6)$ is point 2, making $x_2 = -1$ and $y_2 = 6$ .

STEP 5

So, we have $d = \sqrt{(-1 - (-6))^2 + (6 - (-2))^2}$ .
Let's simplify this: $(-1 - (-6))$ becomes $(-1 + 6) = 5$ , and $(6 - (-2))$ becomes $(6 + 2) = 8$ .
So, $d = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89}$ .
The length of $\overline{JK}$ is $\sqrt{89}$ !

STEP 6

We're going to use our trusty distance formula again to find the length of $\overline{KL}$ .

STEP 7

This time, our points are K at $(-1, 6)$ and L at $(7, 1)$ .
Let's label K as point 1, so $x_1 = -1$ and $y_1 = 6$ , and L as point 2, so $x_2 = 7$ and $y_2 = 1$ .

STEP 8

Plugging into the distance formula, we get $d = \sqrt{(7 - (-1))^2 + (1 - 6)^2}$ .
Simplifying, $(7 - (-1))$ becomes $(7 + 1) = 8$ , and $(1 - 6) = -5$ .
So, $d = \sqrt{8^2 + (-5)^2} = \sqrt{64 + 25} = \sqrt{89}$ .
The length of $\overline{KL}$ is $\sqrt{89}$ !

STEP 9

The length of $\overline{JK}$ is $\sqrt{89}$ , and the length of $\overline{KL}$ is $\sqrt{89}$ .

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Studdy Solution

STEP 1

What is this asking? We need to find the lengths of two sides of a parallelogram on a graph, specifically side JK and a side next to it (either JM or KL). Watch out! Don't mix up the x and y coordinates in the distance formula!
Also, remember we want exact answers, so keep those square roots!

STEP 2

1. Find the length of JK.
2. Find the length of KL.

STEP 3

Alright, let's kick things off with the distance formula!
This formula helps us find the distance between two points on a graph.
Remember, it's like the Pythagorean theorem in disguise!
The distance formula is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

STEP 4

Our dynamic duo of points is J at $(-6, -2)$ and K at $(-1, 6)$ .
Let's plug these coordinates into our super-powered distance formula!
We'll call $(-6, -2)$ point 1, so $x_1 = -6$ and $y_1 = -2$ .
And $(-1, 6)$ is point 2, making $x_2 = -1$ and $y_2 = 6$ .

STEP 5

So, we have $d = \sqrt{(-1 - (-6))^2 + (6 - (-2))^2}$ .
Let's simplify this: $(-1 - (-6))$ becomes $(-1 + 6) = 5$ , and $(6 - (-2))$ becomes $(6 + 2) = 8$ .
So, $d = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89}$ .
The length of $\overline{JK}$ is $\sqrt{89}$ !

STEP 6

We're going to use our trusty distance formula again to find the length of $\overline{KL}$ .

STEP 7

This time, our points are K at $(-1, 6)$ and L at $(7, 1)$ .
Let's label K as point 1, so $x_1 = -1$ and $y_1 = 6$ , and L as point 2, so $x_2 = 7$ and $y_2 = 1$ .

STEP 8

Plugging into the distance formula, we get $d = \sqrt{(7 - (-1))^2 + (1 - 6)^2}$ .
Simplifying, $(7 - (-1))$ becomes $(7 + 1) = 8$ , and $(1 - 6) = -5$ .
So, $d = \sqrt{8^2 + (-5)^2} = \sqrt{64 + 25} = \sqrt{89}$ .
The length of $\overline{KL}$ is $\sqrt{89}$ !

STEP 9

The length of $\overline{JK}$ is $\sqrt{89}$ , and the length of $\overline{KL}$ is $\sqrt{89}$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the length of *JK*.2. Find the length of *KL*.

STEP 3

STEP 4

STEP 5

STEP 6

We're going to use our **trusty** distance formula again to find the length of KL‾\overline{KL}KL.

STEP 7

This time, our points are *K* at (−1,6)(-1, 6)(−1,6) and *L* at (7,1)(7, 1)(7,1).Let's label *K* as point 1, so x1=−1x_1 = -1x1​=−1 and y1=6y_1 = 6y1​=6, and *L* as point 2, so x2=7x_2 = 7x2​=7 and y2=1y_2 = 1y2​=1.

STEP 8

STEP 9

The length of JK‾\overline{JK}JK is 89\sqrt{89}89​, and the length of KL‾\overline{KL}KL is 89\sqrt{89}89​.

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1. Find the length of JK.
2. Find the length of KL.

We're going to use our trusty distance formula again to find the length of $\overline{KL}$ .

This time, our points are K at $(-1, 6)$ and L at $(7, 1)$ .
Let's label K as point 1, so $x_1 = -1$ and $y_1 = 6$ , and L as point 2, so $x_2 = 7$ and $y_2 = 1$ .

The length of $\overline{JK}$ is $\sqrt{89}$ , and the length of $\overline{KL}$ is $\sqrt{89}$ .