Using & Graphing Parallelograms in the Coordinate Plane

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will teach you how to use and graph parallelograms on the coordinate plane. Examples will be provided to help you understand how to graph and use parallelograms.

Parallelograms

Parallelograms are one of the most common shapes found in the real world. Parallelograms are a four sided figures with two pairs of parallel sides.


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Can you think of some real world examples of parallelograms?

The shape of rooms, walls, ceilings, city roads, crop fields, computers, furniture, and many many more are parallelograms. You may think, ok, but aren't some of those rectangles and squares? Well rectangles and squares are parallelograms because they have all of the properties of parallelograms.

There are five properties of parallelograms:

  1. Opposite sides are parallel: This means that the segments have the same slope. The slope of a line is the rate in change of vertical distance over the horizontal distance.
  2. Opposite sides are congruent: Congruent segments are segments with equal length. The sides that are across from each other are congruent.
  3. Opposite angles are congruent: Congruent angles are angles that have equal measures. The angles that are across from each other are congruent.
  4. Consecutive angles are supplementary: The consecutive angles are the angles that are next to each other and they add up to 180 degrees, which means they are supplementary.
  5. Diagonals are bisected lines: The diagonals of a parallelogram are lines that evenly bisect each other, creating two equal sections of each line. There are two diagonals for every parallelogram drawn from one corner to the opposite corner. The diagonals themselves are not congruent, but each half is congruent.

Now that we know all of the properties of parallelograms, we can graph them on the coordinate plane.

Graphing Parallelograms

Example 1: Graph parallelogram ABCD with coordinates A(-7, 5), B(6, 5), C(4, -2) and D(-9, -2), in the coordinate plane. Then show that ABCD is a parallelogram by proving that one pair of sides is both parallel and congruent.

In order to graph the parallelogram, we must know how to graph the coordinates. To graph each point, we first look at the first number in the parentheses, this is the x value. The x value tells us which direction to go, left or right. We move left if the value is negative, and we move right if the value is positive.Next, we look at the second value, this is the y value. The y value tells us whether to move up or down. We move up if it is positive and down if it is negative.

Let's plot these four points. Once we have the four points plotted, we can connect them in order, A to B to C to D, to create our parallelogram.


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There are a few things to notice. If you look at segments AB and CD, do you notice that they are both horizontal lines? Horizontal lines have the same slope of 0 since these two sides are both parallel. We can also count the spaces between A and B, and C and D to see if they are the same length.

If we count AB, we get 13 spaces. If we count CD, we also get 13 spaces. This means that these two sides are both parallel and congruent, which proves it is a parallelogram.

Example 2: Graph parallelogram EFGH with coordinates E(-1, 5), F(2, 8), G(4, 4) and H(1, 1). Then show that EFGH is a parallelogram by proving that opposite sides are parallel.

First, we plot the points, and then connect them in order.


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The next step is to figure out the slope or the rate of change of the vertical distance over the horizontal distance. The easiest way to determine the slope on the graph is to count. It is also known as the rise over the run, the shorthand way to remember how to calculate the slope.

Let's start with side EF. Notice, when looking at the line from left to right, the line increases. This means it will have a positive slope.

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