# 6.6: Reflections and Symmetry

Difficulty Level: At Grade Created by: CK-12

## Introduction

Building the Teepee

Dylan looked at the work his friend Marcus was doing in class. Marcus had decided to design and build a tepee. Like Dylan’s geodesic dome, Marcus was having a difficult time with the construction aspect of the teepee.

“What seems to be the trouble?” Dylan asked Marcus as he saw Marcus sit down frustrated next to his sticks and cloth covering.

“This thing won’t stand up. I put the sticks together. They are all the same length, then I try to put the canvas over the sticks and it doesn’t fit. I am so frustrated!” Marcus exclaimed putting his head in his hands.

Dylan looked at the sticks and then as the canvas. As soon as he looked at the canvas, he knew what was wrong with Marcus’ design.

“I know how to fix it.”

“How?” Marcus asked puzzled.

“Symmetry is the key here, not the length of the sticks.” Dylan said.

Do you know what Dylan means? What is symmetry? How does a teepee have symmetry? What would Marcus have to do to be sure that his canvas was symmetrical?

Pay attention during this lesson and you will learn all that you need to know to solve this problem and help Marcus with his teepee.

What You Will Learn

In this lesson you will learn how to complete the following skills.

• Recognize reflection transformations (flips) and lines of reflection.
• Use coordinate rotation to describe reflections, given figures and their resulting images reflected in either the \begin{align*}y-\end{align*}axis or the \begin{align*}x-\end{align*}axis.
• Use coordinate notation to reflect given figures in either or both axes.
• Identify lines of symmetry in real – world objects.

Teaching Time

I. Recognize Reflection Transformations (Flips) and Lines of Reflection

In the last lesson, you learned how to determine whether or not two polygons were congruent. When you did this, you were given two polygons to work with. In this lesson, we are going to create congruent figures by using transformations. A transformation is moved in some way. One kind of transformation is called a reflection or a flip.

To understand transformations, we first need to review the coordinate plane. We examine and perform reflections in the coordinate plane. The coordinate plane is a representation of two-dimensional space. It has a horizontal axis, called the \begin{align*}x-\end{align*}axis, and a vertical axis, called the \begin{align*}y-\end{align*}axis.

We can graph and move geometric figures on the coordinate plane. Here is a picture of the coordinate plane.

When we work with reflections or flips, we can see a figure in the coordinate plane. Look at this example.

Here are two right triangles. We can say that they are reflected over the \begin{align*}y-\end{align*}axis because the \begin{align*}y-\end{align*}axis is acting like a mirror for the two triangles. We call this the line of reflection, because the \begin{align*}y-\end{align*}axis is doing the reflecting. Imagine standing in front of a mirror and holding up your left hand. Where is your hand in the mirror’s reflection? A reflected figure works the same way: when we flip it over the line of reflection, all of its points are reversed.

We can reflect an image over the \begin{align*}x-\end{align*}axis or over the \begin{align*}y-\end{align*}axis.

II. Use Coordinate Notation to Describe Reflections, Given Figures and Their Resulting Images

We can draw reflections on the coordinate plane, and we can also write about the reflections that we draw using something called coordinate notation.

Think back to when you plotted points on the coordinate plane.

Look at this example.

Here point \begin{align*}A\end{align*} is plotted on the coordinate plane. This is a drawing of the point plotted. We can also write about it being plotted. To write about it, we name it with a set of ordered point. We write the \begin{align*}x-\end{align*}coordinate first and then the \begin{align*}y-\end{align*}coordinate.

Point \begin{align*}A = (3, 4)\end{align*}

This is an example of coordinate notation.

When a figure is drawn on the coordinate plane, we can use coordinate notation to describe the figure drawn. If we draw a triangle, then there will be three sets of ordered pairs to represent each vertex that has been drawn.

Look at this example.

This triangle has three vertices that represent it.

(-1, 1)

(-3, 1)

(-1, 6)

What if we reflected this triangle over the \begin{align*}y-\end{align*}axis?

If we did this, then the coordinates of the first triangle would change. Let’s look at this reflection and examine the new coordinates. As with all things in math, look for a pattern first.

The reflected triangle has the following coordinates for vertices.

(1, 1)

(3, 1)

(1, 6)

Do you see any patterns?

If you look carefully, you will see that the \begin{align*}x-\end{align*}coordinates of the reflected triangle are opposite those of the first triangle. This is a rule to help you.

Write these two rules down in your notebooks.

III. Use Coordinate Notation to Reflect Given Figures in Either or Both Axes

Now that you know the two rules for figuring out the coordinates of a figure reflected on the coordinate plane, so you can use those rules to figure out new reflections whether you have been given an image or not.

Let’s look at an example.

Example

What would be the new coordinates of a figure reflected over the \begin{align*}x-\end{align*}axis?

Now, we can look at this figure and write down the coordinates of this trapezoid.

(2, 1)

(7, 1)

(3, 3)

(6, 3)

Next, we can use the rule to figure out the new coordinates of the reflected trapezoid. We are going to reflect this trapezoid over the \begin{align*}x-\end{align*}axis, so the \begin{align*}y-\end{align*}coordinates will become the opposite and the \begin{align*}x-\end{align*}coordinates will remain the same. Here are the new coordinates.

(2, -1)

(7, -1)

(3, -3)

(6, -3)

Now we can graph the reflected trapezoid on the coordinate plane.

You can see that the \begin{align*}x-\end{align*}axis forms a line of reflection so that one trapezoid becomes the mirror image of the other trapezoid.

IV. Identify Lines of Symmetry in Real – World Objects

Throughout this lesson you have learned how to reflect figures on the coordinate plane. When this happens, we can see a mirror of two figures. We reflected figures over the \begin{align*}x-\end{align*}axis and over the \begin{align*}y-\end{align*}axis. Sometimes, a figure will have parts that mirror themselves within one object. In this case, parts of the object match other parts of the picture. This is called symmetry. Let’s look at an example.

Look at this heart. It has two sides that match. The heart is symmetrical because there is symmetry in its design. This heart can be divided in half vertically where one half matches the other half. This line that divides the heart into matching parts is called the line of symmetry.

We can determine other lines of symmetry by looking at other objects.

Look at this cross. It has two lines of symmetry. If you look, the cross can be divided in half perfectly vertically and in half horizontally. This means that there are two lines of symmetry in the cross.

We can find symmetry all around us. There is symmetry in real – world objects that we see all the time. Look around you today and locate three different things that have lines of symmetry.

Here are some butterflies to think about.

## Real-Life Example Completed

Building the Teepee

Here is the original problem from the introduction. Reread it and then answer the questions at the end of the problem.

Dylan looked at the work his friend Marcus was doing in class. Marcus had decided to design and build a tepee. Like Dylan’s geodesic dome, Marcus was having a difficult time with the construction aspect of the teepee.

“What seems to be the trouble?” Dylan asked Marcus as he saw Marcus sit down frustrated next to his sticks and cloth covering.

“This thing won’t stand up. I put the sticks together. They are all the same length, then I try to put the canvas over the sticks and it doesn’t fit. I am so frustrated!” Marcus exclaimed putting his head in his hands.

Dylan looked at the sticks and then as the canvas. As soon as he looked at the canvas, he knew what was wrong with Marcus’ design.

“I know how to fix it.”

“How?” Marcus asked puzzled.

“Symmetry is the key here, not the length of the sticks.” Dylan said.

Do you know what Dylan means? What is symmetry? How does a teepee have symmetry? What would Marcus have to do to be sure that his canvas was symmetrical?

Answer these questions before continuing.

Solution to Real – Life Example

Here are the three questions posed at the end of the problem in the introduction. Reread them and then answer them.

What is symmetry?

Symmetry is when two halves of an object match. In other words, you can divide the object into parts and the parts are congruent. A heart is a symmetrical object, so is a teepee.

How does a teepee have symmetry?

A teepee has symmetry because it can be divided in half so that one half of the teepee matches the other half.

What would Marcus have to do to be sure that his canvas was symmetrical?

While Marcus was sure that his sticks were all the same length that is only half of the necessary piece. Marcus also needs to be sure that the canvas is the same all the way around. If he does, then all sides will match or be symmetrical, if not then one side will be different that the other.

## Vocabulary

Here are the vocabulary words found in this lesson.

Transformation
a way of shifting or moving a geometric figure on the coordinate plane.
Coordinate Plane
a representation of two – dimensional space with an \begin{align*}x-\end{align*}axis, a \begin{align*}y-\end{align*}axis and a coordinates.
Reflection
a transformation known as a flip where a mirror image of a figure is created.
Line of Reflection
the \begin{align*}x\end{align*} or \begin{align*}y-\end{align*}axis which is the mirror for the reflected figure on the coordinate plane.
Coordinate Notation
using ordered pairs to represent the vertices of a figure on the coordinate plane.
Symmetry
when an object has the ability to be divided into matching parts.
Line of Symmetry
the line that divides an object into matching parts.

## Time to Practice

Directions: Define the following terms.

1. Reflection
2. Coordinate Plane
3. \begin{align*}\underline{x-}\end{align*}axis
4. \begin{align*}\underline{y-}\end{align*}axis

Directions: Write each set of coordinates for a reflection of each figure over the \begin{align*}x-\end{align*}axis.

1. (1,3) (2,5) (3, 2)
2. (2, 1) (5, 1) (2, 4)
3. (-1, 1) (-1, 3) (-4, 1)
4. (1, 2) (1, 5) (5, 2) (5, 5)
5. (1, 2) (6, 1) (6, 3) (2, 3)
6. (-1, 3) (-3, 1) (-5, 1) (-4, 6)

Directions: Write a new series of coordinates for a figure reflected over the \begin{align*}y-\end{align*}axis.

1. (1, 3) (2, 5) (3, 2)
2. (-1, 1) (-1, 3) (-4, 1)
3. (2, 1) (5, 1) (2, 4)
4. (1, 2) (1, 5) (5, 2) (5, 5)
5. (-1, 3) (-3, 1) (-5, 1) (-4, 6)

Directions: Now go back and identify each figure that was reflected.

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