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# Reflections

## Transformations that turn a figure into its mirror image by flipping it over a line.

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Recognizing Reflections

Kevin and his sister Kim went to visit their great aunt in the country. On the drive in, they 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’s a reflection!” Kim exclaimed. Kevin is not sure what this means, how can Kim explain it to him?

In this concept, you will learn to recognize reflections.

### Reflections

Congruent figures are created by using transformations. A transformation is a move in some way. One kind of transformation is called a reflection or a flip.

Look at the figures below. Each of the figures can be reflected over a line of reflection.

You can also find reflections on the coordinate plane.

To understand reflection transformations, a review of the coordinate plane is necessary. A reflection performed on the coordinate plane is provides a visual representation of the transformation. 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.

Geometric figures can be graphed and moved on the coordinate plane. Here is a picture of the coordinate plane.

Look at the image below.

The image on the right is the pre-image or the triangle to which the transformation will be applied. This triangle has been 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. This line that acts like the mirror when a reflection occurs is called the line of reflection. The triangle on the left is the image or the result of the triangle on the right being reflected over the y\begin{align*}y\end{align*}-axis.

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 you flip it over the line of reflection, all of its points are reversed.

You can reflect an image over the x\begin{align*}x\end{align*}-axis or over the y\begin{align*}y\end{align*}-axis. These two lines of reflection will be used to perform reflections on the coordinate plane. The reflections performed on the coordinate plane can be described using coordinate notation.

Let’s take a look at how to express a reflection using coordinate notation.

The point A is plotted on the coordinate plane.

To write about point A\begin{align*}A\end{align*} being plotted, name it using an ordered pair to represent both its x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}-coordinates. An ordered pair is written in the form (x,y)\begin{align*}(x, y)\end{align*} where x\begin{align*}x\end{align*} is the x\begin{align*}x\end{align*}-coordinate and y\begin{align*}y\end{align*} is the y\begin{align*}y\end{align*}-coordinate of the plotted point.

Therefore, Point A=(3,4)\begin{align*}A = (3, 4)\end{align*} has been represented using coordinate notation.

When a figure is drawn on the coordinate plane, coordinate notation can be used to describe the figure drawn. If a triangle is drawn on a coordinate plane, then three sets of ordered pairs must be written to represent each vertex of the triangle.

Let’s look at an example.

This triangle has three vertices that can be named using coordinate notation as:

(-1, 1), (-3, 1), and (-1, 6).

If this triangle were reflected over the y\begin{align*}y\end{align*}-axis, what coordinate notation would be used to name its vertices?

The diagram below shows the preimage triangle on the left and the image triangle on the right.

The image triangle (the one on the right side) has the following coordinates for vertices.

(1, 1), (3, 1) and (1, 6)

Let’s compare the coordinates of the two triangles side by side to see if there is a pattern.

Triangle1Triangle2(left side - preimage)  (right side - image)(1,1)(1,1)(3,1)(3,1)(1,6)(1,6)\begin{align*}\begin{array}{rcl} && \text{Triangle} - 1 \qquad \qquad \qquad \quad \quad \text{Triangle} - 2 \\ && \text{(left side - preimage)}\qquad \qquad \ \ \text{(right side - image)} \\ && (-1, 1) \qquad \qquad \qquad \qquad \qquad (1, 1) \\ && (-3, 1) \qquad \qquad \qquad \qquad \qquad (3, 1) \\ && (-1, 6) \qquad \qquad \qquad \qquad \qquad (1, 6) \\ \end{array}\end{align*}

You can see that the x\begin{align*}x\end{align*}-coordinates of the reflected triangle are opposite those of the first triangle.

Remember, when figure is reflected over the y\begin{align*}y\end{align*} axis, the x\begin{align*}x\end{align*} coordinates are opposite in the reflection. When a figure is reflected over the x\begin{align*}x\end{align*} axis, the y\begin{align*}y\end{align*} coordinates are opposite in the reflection.

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.

### Examples

#### Example 1

Earlier, you were given a problem about Kim’s reflection.

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 the reflection would be perfect.

#### Example 2

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

First, look at this figure and write down the coordinates of this trapezoid.

(2, 1)

(7, 1)

(3, 3)

(6, 3)

Next, use the rule to figure out the new coordinates of the reflected trapezoid. This trapezoid is being reflected over the x\begin{align*}x\end{align*}-axis, so the y\begin{align*}y\end{align*}-coordinates will become the opposite while the x\begin{align*}x\end{align*}-coordinates remain the same.

The new coordinates are:

(2, -1)

(7, -1)

(3, -3)

(6, -3)

Then, graph the reflected trapezoid on the coordinate plane.

This graph shows the reflection.

The x\begin{align*}x\end{align*}-axis forms a line of reflection so that the trapezoid below the x\begin{align*}x\end{align*}-axis is the mirror image of the trapezoid above the x\begin{align*}x\end{align*}-axis.

#### Example 3

Is this an example of a reflection?

First, determine if the figure can be divided into sides that are perfectly matched.

Look at the orange slice. It looks like a line of symmetry can be drawn in four parts of the diagram to create mirror images.

Next, draw the lines of symmetry to see if this figure is an example of a reflection.

The figure can be divided so that one side perfectly matches the other.

#### Example 4

Is this an example of a reflection?

First, determine if the shape can be divided into sides that are perfectly matched.

It does not look like a line of symmetry can be drawn for this diagram to create mirror images.

### Review

Define the following terms.

1. Reflection

2. Coordinate Plane

3. X\begin{align*}X\end{align*}-axis

4. Y\begin{align*}Y\end{align*}-axis

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

5. (1, 3) (2, 5) (3, 2)

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

7. (-1, 1) (-1, 3) (-4, 1)

8. (1, 2) (1, 5) (5, 2) (5, 5)

9. (1, 2) (6, 1) (6, 3) (2, 3)

10. (-1, 3) (-3, 1) (-5, 1) (-4, 6)

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

11. (1, 3) (2, 5) (3, 2)

12. (-1, 1) (-1, 3) (-4, 1)

13. (2, 1) (5, 1) (2, 4)

14. (1, 2) (1, 5) (5, 2) (5, 5)

15. (-1, 3) (-3, 1) (-5, 1) (-4, 6)

### 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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