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# Reflections

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

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Reflections

Reflections are often informally called “flips”. Why is this?

#### Guidance

A reflection is one example of a rigid transformation. A reflection across line \begin{align*}l\end{align*} moves each point \begin{align*}P\end{align*} to \begin{align*}P^\prime\end{align*}  such that line \begin{align*}l\end{align*} is the perpendicular bisector of the segment connecting \begin{align*}P\end{align*} and \begin{align*}P^\prime\end{align*}. Below, the quadrilateral has been reflected across line \begin{align*}l\end{align*} to create a new quadrilateral (the image).

Reflections move all points of a shape across a line called the line of reflection. A point and its corresponding point in the image are each the same distance from the line.

Notice that the segments connecting each point with its corresponding point are all perpendicular to line \begin{align*}l\end{align*}. Because each point and its corresponding point are the same distance from the line, line \begin{align*}l\end{align*} bisects each of these segments.  This is why line \begin{align*}l\end{align*} is called the perpendicular bisector for each of these segments.

Keep in mind that you can perform reflections even when the line of reflection is “slanted” or the grid is not visible; however, it is much harder to do by hand.

Example A

Describe the line of reflection that created the reflected image below.

Solution: The fact that point \begin{align*}D\end{align*} is in the same location as point \begin{align*}D^\prime\end{align*} tells you that the line of reflection passes through point \begin{align*}D\end{align*}. Imagine folding the graph so that each point on the original parallelogram matched its point on the image. Where would the fold be?

The line of reflection is the line \begin{align*}y=x\end{align*}. When reflections are performed on graph paper with axes, you can define the lines of reflections with their equations.

Example B

Is the following transformation a reflection?

Solution: Even though overall both the parallelogram and its image are 2 units from the \begin{align*}x\end{align*}-axis, each individual point and its image are not the same distance from the \begin{align*}x\end{align*}-axis. For example, point \begin{align*}A\end{align*} is 3 units from the \begin{align*}x\end{align*}-axis and point \begin{align*}A^\prime\end{align*} is 2 units from the \begin{align*}x\end{align*}-axis. The line of reflection for points \begin{align*}A\end{align*} and \begin{align*}A^\prime\end{align*} would be the line \begin{align*}y=\frac{1}{2}\end{align*}, which is not the same line of reflection for points \begin{align*}D\end{align*} and \begin{align*}D^\prime\end{align*}. This is not a reflection (it's a translation).

Example C

Perform the reflection across line \begin{align*}l\end{align*}.

Solution: Draw a perpendicular line from each point that defines the parallelogram to line \begin{align*}l\end{align*}. Count how many units there are between each point and line \begin{align*}l\end{align*} along the perpendicular lines. Count the same number of units on the other side of line \begin{align*}l\end{align*} along the perpendicular lines to create the image.

Concept Problem Revisited

A reflection is informally called a “flip” because it's as if you are flipping the shape over the line of reflection.

#### Vocabulary

A reflection across line \begin{align*}l\end{align*} moves each point \begin{align*}P\end{align*} to \begin{align*}P^\prime\end{align*} such that line \begin{align*}l\end{align*} is the perpendicular bisector of the segment connecting \begin{align*}P\end{align*} and \begin{align*}P^\prime\end{align*}. An informal way to think about a reflection is as a “flip”.

A rigid transformation is a transformation that preserves distance and angles.

A perpendicular bisector is a line that bisects a segment (cuts it in half) and is perpendicular to the segment.

#### Guided Practice

1. Reflect the triangle across the \begin{align*}x\end{align*}-axis.

2. From #1, what do you notice about each point and its image when a reflection is done across the \begin{align*}x\end{align*}-axis?

3. Describe the line of reflection that created the reflected image below.

1.

2. The \begin{align*}x\end{align*}-coordinate of each point and its image are the same, but the \begin{align*}y\end{align*}-coordinate has changed sign. You could describe this as \begin{align*}(x, y) \rightarrow (x, -y)\end{align*}.

3. Connect \begin{align*}A\end{align*} with \begin{align*}A^\prime\end{align*}. This line segment has a slope of -1 and a midpoint at (2, 1). The line of reflection is the perpendicular bisector of this segment. This means it passes through its midpoint and has an opposite reciprocal slope \begin{align*}\left(-\frac{1}{1} \rightarrow +\frac{1}{1}\right) \end{align*}. The line of reflection is the line \begin{align*}y=x-1\end{align*}.

#### Practice

Describe the line of reflection that created each of the reflected images below.

1.

2.

3.

4.

Reflect each shape across line \begin{align*}l\end{align*}.

5.

6.

7.

8.

Is the transformation a reflection? Explain.

9.

10.

11. Reflect a shape across the \begin{align*}y\end{align*}-axis. How are the points of the original shape related to the points of the image?

12. The point (7, 2) is reflected across the \begin{align*}y\end{align*}-axis. Can you find the coordinates of the image point using the relationship you found in #11?

13. Reflect a shape across the line \begin{align*}y=x\end{align*}. How are the points of the original shape related to the points of the image?

14. The point (7, 2) is reflected across the line \begin{align*}y=x\end{align*}. Can you find the coordinates of the image point using the relationship you found in #13?

15. Reflect a shape across the line \begin{align*}y=-x\end{align*}. How are the points of the original shape related to the points of the image?

16. The point (7, 2) is reflected across the line \begin{align*}y=-x\end{align*}. Can you find the coordinates of the image point using the relationship you found in #15?

### Vocabulary Language: English

Coordinate Plane

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

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

Image

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

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

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

Preimage

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

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

Transformation

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

Rigid Transformation

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