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# 6.12: Recognizing Reflections

Difficulty Level: At Grade Created by: CK-12
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Have you ever been to an estate in the country? Take a look at this dilemma.

Kevin and his sister Kim went to visit their great aunt in the country. On the drive in, Kevin and Kim enjoyed looking at the scenery. It was so different from their home in the city. As they turned down the drive leading up to the estate, Kim gasped at the lovely tree lined road.

"Look at the symmetry. It is a perfect reflection," she said.

Do you know what she means?

### Guidance

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.

We can look at these figures and see that they can be reflected over a line of reflection.

We can also find reflections on the coordinate plane.

To understand reflection 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 x\begin{align*}x-\end{align*}axis, and a vertical axis, called the y\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 one.

Here are two right triangles. We can say that they are reflected over the y\begin{align*}y-\end{align*}axis because the y\begin{align*}y-\end{align*}axis is acting like a mirror for the two triangles. We call this the line of reflection, because the y\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 x\begin{align*}x-\end{align*}axis or over the y\begin{align*}y-\end{align*}axis.

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.

Take a look at how this is done.

Here point A\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 x\begin{align*}x-\end{align*}coordinate first and then the y\begin{align*}y-\end{align*}coordinate.

Point A=(3,4)\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 one.

This triangle has three vertices that represent it.

(-1, 1)

(-3, 1)

(-1, 6)

What if we reflected this triangle over the y\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 x\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.

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.

#### Example A

Define reflection.

Solution: A reflection is a perfect mirror image.

#### Example B

Is this an example of a reflection?

Solution: Yes, because the figure can be divided so that one side perfectly matches the other.

#### Example C

Is this an example of a reflection?

Solution: No, because there are specific images on the circle that would not reflect if the image was divided.

Now let's go back to the dilemma from the beginning of the Concept.

Kim made the statement that she did because one side of the road is a perfect reflection of the other side. In other words, one side matches the other side. You could draw a line right down the center of the road separating the left side from the right side and then the reflection would be perfect.

### Vocabulary

Transformation
a way of shifting or moving a geometric figure on the coordinate plane.
Coordinate Plane
a representation of two – dimensional space with an x\begin{align*}x-\end{align*}axis, a y\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 x\begin{align*}x\end{align*} or y\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.

### Guided Practice

Here is one for you to try on your own.

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

Solution

First, 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 x\begin{align*}x-\end{align*}axis, so the y\begin{align*}y-\end{align*}coordinates will become the opposite and the x\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 x\begin{align*}x-\end{align*}axis forms a line of reflection so that one trapezoid becomes the mirror image of the other trapezoid.

### Practice

Directions: Define the following terms.

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

Directions: Write each set of coordinates for a reflection of each figure over the x\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 y\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)

### My Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

Coordinate Notation

A coordinate point is the description of a location on the coordinate plane. Coordinate points are commonly written in the form (x, y) where x is the horizontal distance from the origin, and y is the vertical distance from the origin.

Coordinate Plane

The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. The coordinate plane is also called a Cartesian Plane.

Geometric Patterns

Geometric patterns are visual patterns of geometric figures that follow a rule.

Image

The image is the final appearance of a figure after a transformation operation.

perpendicular bisector

A perpendicular bisector of a line segment passes through the midpoint of the line segment and intersects the line segment at $90^\circ$.

Perpendicular lines

Perpendicular lines are lines that intersect at a $90^{\circ}$ angle.

Preimage

The pre-image is the original appearance of a figure in a transformation operation.

Reflection

A reflection is a transformation that flips a figure on the coordinate plane across a given line without changing the shape or size of the figure.

Transformation

A transformation moves a figure in some way on the coordinate plane.

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.

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