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How to Find the Perimeter of a Parallelogram in the Coordinate Plane

Step 1: Determine the vertices of the parallelogram.

Step 2: Use the distance formula to determine the length of two adjacent sides. (If a side is horizontal or vertical you can determine the length without using the distance formula!)

Step 3: Multiply each of the distances calculated in the previous step by 2 and add the results together.

How to Find the Perimeter of a Parallelogram in the Coordinate Plane: Vocabulary and Formula

Perimeter: Perimeter is the distance around the outside of a given two-dimensional shape or area.

Parallelogram: A parallelogram is a 4 sided figure with opposite sides of equal length and parallel to each other.

Adjacent sides: In a figure, adjacent sides are sides that meet at a vertex.

Distance Formula: The distance formula allows us to calculate the distance between two coordinate points on the cartesian plane.

• Given to points on the plane {eq}(x_1, y_1), (x_2, y_2) {/eq} the distance between the two points can be calculated as follows:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Let's try two example problems to learn how to find the perimeter of a parallelogram in the coordinate plane.

How to Find the Perimeter of a Parallelogram in the Coordinate Plane: Example 1

Use the given image to determine the perimeter of parallelogram ABCD.

Step 1: Determine the vertices of the parallelogram.

Using the diagram we find the following vertices:

$$\begin{align} A&(-4,-6)\\ B&(4,-6)\\ C&(6,-2)\\ D&(-2,-2) \end{align} $$

Step 2: Use the distance formula to determine the length of two adjacent sides. (If a side is horizontal or vertical you can determine the length without using the distance formula!)

Given that we've been told the figure is a parallelogram and that opposite sides of a parallelogram are equal, we need only calculate the distance for two adjacent sides to know the length of all sides.

Sides AB and DC are opposite and therefore parallel and the same length. Notice that these sides run horizontally along the coordinate plane, which means we don't need the distance formula to calculate the length of these matching sides. We can count the squares between A and B, which gives us 8 units. However, this is not always practical for larger values.

We could also use the distance formula for AB but all we really need to find the distance between two points on a horizontal or vertical line, is to take the absolute value of the difference between:

• the {eq}x {/eq} values for a horizontal line (note that {eq}y {/eq} values would be identical).

• the {eq}y {/eq} values for a vertical line (note that the {eq}x {/eq} values would be identical).

For our horizontal line, we have the difference between the two {eq}x {/eq} values of A and B.

$$\begin{align} d_{AB}&=\left |-4 -4 \right |\\ &=\left |-8 \right |\\ &=8 \end{align} $$

Thus, AB = 8 units.

Now, for the other two sides, AD and BC. Again, since this is a parallelogram, they are parallel to each other and of equal length. So, let's use the distance formula to calculate one of the sides, AD.

$$\begin{align} &A(-4,-6) \text{ and } D(-2,-2)\\ d_{AD} &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\ &= \sqrt{(-2 - (-4))^2 + (-2 - (-6))^2}\\ &= \sqrt{(-2 + 4)^2 + (-2+ 6)^2}\\ &= \sqrt{(2)^2 + (4)^2}\\ &= \sqrt{4 + 16}\\ &= \sqrt{20}\\ \end{align} $$

Thus, AD = {eq}\sqrt{20} {/eq} units.

Step 3: Multiply each of the distances calculated in the previous step by 2 and add the results together.

Since AB is 8 units then DC is 8 units.

Since AD is {eq}\sqrt{20} {/eq} units then BC is {eq}\sqrt{20} {/eq} units.

So, we need only multiply each length by two to account for the two sides of the same length and add it all up!

$$\begin{align} \text{Perimeter } &= 2AB + 2AD\\ &= 2(8) + 2\sqrt{20}\\ &=16 + 2\sqrt{20}\\ &\approx 24.94 \end{align} $$

Therefore, the perimeter of the parallelogram ABCD is 24.94 units.

Tip: If we had not been told the shape is a parallelogram, we can not assume the opposite sides are of equal length and would have had to calculate all 4 sides to be certain of the side lengths!

How to Find the Perimeter of a Parallelogram in the Coordinate Plane: Example 2

Use the given image to determine the perimeter of parallelogram PQRS.

Step 1: Determine the vertices of the parallelogram.

$$\begin{align} P&(1,-1)\\ Q&(-3,4)\\ R&(-1,5)\\ S&(3,0)\\ \end{align} $$

Step 2: Use the distance formula to determine the length of two adjacent sides. (If a side is horizontal or vertical you can determine the length without using the distance formula!)

Since there are no horizontal or vertical lines, we'll use the distance formula to calculate adjacent sides PS and PQ.

$$\begin{align} &P(1,-1) \text{ and } S(3,0) && P(1,-1)\text{ and } Q(-3,4)\\ d_{PS} &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} & d_{PQ} &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\ &= \sqrt{(3 - (1))^2 + (0 - (-1))^2} &&= \sqrt{(-3 - (1))^2 + (4 - (-1))^2}\\ &= \sqrt{(2)^2 + (1)^2} &&= \sqrt{(-4)^2 + (5)^2}\\ &= \sqrt{4+ 1} &&= \sqrt{16 + 25}\\ &= \sqrt{5} &&= \sqrt{41}\\ \end{align} $$

Step 3: Multiply each of the distances calculated in the previous step by 2 and add the results together.

$$\begin{align} \text{Perimeter } &= 2PS + 2PQ\\ &= 2(\sqrt{5}) + 2(\sqrt{41})\\ &\approx 17.28 \end{align} $$

Therefore, the perimeter of the parallelogram PQRS is 17.28 units.

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