6.6: Reflections and Symmetry
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*}
y− axis or the \begin{align*}x\end{align*}x− 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 twodimensional space. It has a horizontal axis, called the \begin{align*}x\end{align*}
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*}
We can reflect an image over the \begin{align*}x\end{align*}
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*}
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*}
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*}
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*}
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*}
(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*}
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*}
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.
RealLife 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*}
x− axis, a \begin{align*}y\end{align*}y− 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*}
x or \begin{align*}y\end{align*}y− 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.
 Reflection
 Coordinate Plane

\begin{align*}\underline{x}\end{align*}
x−−−− axis  \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.
 (1,3) (2,5) (3, 2)
 (2, 1) (5, 1) (2, 4)
 (1, 1) (1, 3) (4, 1)
 (1, 2) (1, 5) (5, 2) (5, 5)
 (1, 2) (6, 1) (6, 3) (2, 3)
 (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, 3) (2, 5) (3, 2)
 (1, 1) (1, 3) (4, 1)
 (2, 1) (5, 1) (2, 4)
 (1, 2) (1, 5) (5, 2) (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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