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Families of Lines

Lines sharing a point or slope

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Parallel & Perpendicular Lines
Teacher Contributed

Real World Applications – Algebra I


Where do we see parallel and perpendicular lines in every day life?

Student Exploration

Things you might need: graph paper, ruler, markers

You will need a printer to print pictures, or you might need some expert Photoshop skills for this activity!

You’ll first need to look around your neighborhood and find examples of parallel and perpendicular lines. Once you get pictures, you will then create equations for all of these lines you’ve found. Let’s do one of these together, then you try on your own.

I found the following picture of a car grille. Do you think this represents a set of parallel or perpendicular lines?

This image shows sets of parallel lines. I’m going to add a set of \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axes so we can figure out the equations for both of these lines. After drawing the set of axes, we all have to make sure that the tick marks are all the same. See below.

To find the equation of the first line (let’s say it’s the leftmost line drawn in turquoise below), we need to calculate the slope and find the \begin{align*}y-\end{align*}intercept.

Let’s calculate the slope first. I’m going to find two different points on this line and find the difference between the \begin{align*}y-\end{align*}values and divide it by the difference of the \begin{align*}x-\end{align*}values. To find the \begin{align*}y-\end{align*}intercept on a graph, it’s easy! I’ll just look at where the line intersects the \begin{align*}y-\end{align*}axis.

For the turquoise line, it looks like the slope is \begin{align*}\frac{4}{1}\end{align*}, or 4. I looked at the points at the \begin{align*}y-\end{align*}intercept, which is at (0, 1), and (1, 5). So, the equation for this line is \begin{align*}y = 4x + 1\end{align*}.

We will also do the same for the purple line. The \begin{align*}y-\end{align*}intercept looks like it’s at (0, -11). To find the slope, I can look at the point between the \begin{align*}y-\end{align*}intercept and (1, -7). So, the equation for this line is \begin{align*}y = 4x - 11\end{align*}.

Now, we should notice that parallel lines have the same slope, and different \begin{align*}y-\end{align*}intercepts. Do we see that represented in our equations?

What are other equations that you can see on the car grille? This car grille represents one family of functions, where the slopes of all of the lines are the same.

Now, let’s take a look at this picture below of kitchen tiles.

I took a piece of this picture and added some axes and perpendicular lines.

Let’s find the equations of these lines and verify that these are perpendicular lines.

First, let’s look at the turquoise line. It looks like the \begin{align*}y-\end{align*}intercept is at (0, 1), and the slope is \begin{align*}\frac{1}{1}\end{align*} or 1. The equation is then, \begin{align*}y = x + 1\end{align*}.

For the purple line, the \begin{align*}y-\end{align*}intercept is at (0, 7), and the slope is \begin{align*}\frac{-1}{1}\end{align*}, or -1. The equation is \begin{align*}y = -x + 7\end{align*}.

Looking at the two equations, we can tell that this represents a pair of perpendicular lines because the slopes are opposite reciprocals of each other. Remember, the \begin{align*}y-\end{align*}intercepts also have to be different.

Extension Investigation

Try finding a picture online and print it out, or take a picture and get it developed of both parallel lines and a picture that represents perpendicular lines. Find the equations of these lines, and prove that they are parallel or perpendicular using Algebra!

Also try to find a picture that would represent the second family of functions, where the y-intercept is the same for all of the lines. What are the equations of those lines?

Resources Cited

Shutterstock images: 48660352, 56409679

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