# 10.5: Translating and Reflecting

## Learning Objectives

- Graph a
*translation*in a coordinate plane. - Recognize that a
*translation*is an*isometry.* - Find the
*reflection*of a point in a line on a coordinate plane. - Verify that a
*reflection*is an*isometry.*

## Translations

A **translation** moves *every* point a given *horizontal* distance and/or a given *vertical* distance.

- When a point is moved a certain distance horizontally and/or vertically, the move is called a ______________________________.

For example, if a **translation** moves point 2 units to the *right* and 4 units *up* to , then this **translation** moves *every* point in a larger figure the *same way.*

*The symbol next to the letter above is called the* *prime**symbol.*

*The* *prime**symbol looks like an apostrophe like you may use to show possessive, such as, “that is my brother’s book.”*

*(The apostrophe is before the s in brother’s)*

*In math, we use the* *prime**symbol to show that two things are related.*

*In the* *translation**above, the original point is related to the translated point, so instead of renaming the translated point, we use the* *prime**symbol to show this.*

The *original* point (or figure) is called the **preimage** and the *translated* point (or figure) is called the **image.** In the example given above, the **preimage** is point and the **image** is point . The **image** is *designated* (or *shown*) with the **prime** symbol.

- Another name for the
*original*point is the __________________________. - Another name for the
*translated*point is the __________________________. - The
*translated*point uses the _______________ symbol next to its naming letter.

**Example 1**

*The point in a* *translation**becomes the point . What is the* *image**of* *in the same* *translation**?*

Point moved 1 unit to the *left* and 3 units *down* to get to . Point will also move 1 unit to the *left* and 3 units *down*.

We *subtract* 1 from the coordinate and 3 from the coordinate of point :

is the **image** of .

Using the **Distance Formula,** you can notice the following:

Since the endpoints of and moved the *same* distance horizontally and vertically, both segments have the *same length.*

## Translation is an Isometry

An **isometry** is a transformation in which *distance* is “preserved.” This means that the *distance* between any two points in the **preimage** (*before* the **translation**) is the *same* as the *distance* between the points in the **image** (*after* the **translation**).

- An
**isometry**is when ______________________________ is*preserved*from the**preimage**to the**image.**

As you saw in Example 1 above:

The **preimage** the **image** (since they are both equal to )

Would we get the same result for any other point in this translation? The answer is yes. It is clear that for any point , the distance from to will be . Every point moves units to its **image.**

This is true in general:

**Translation Isometry Theorem**

Every **translation** in the coordinate plane is an **isometry.**

- Every translation in an coordinate plane is an ________________________.

## Reflection in a Line

A **reflection** in a line is as if the line were a *mirror:*

- When an object is
**reflected**in a line, the line is like a ______________________.

An object **reflects** in the mirror, and we see the **image** of the object.

- The
**image**is the*same*distance behind the mirror line as the object is in front of the mirror line. - The “line of sight” from the
*object*to the*mirror*is**perpendicular**to the mirror line itself. - The “line of sight” from the
*image*to the*mirror*is also**perpendicular**to the mirror line.

## Reflection of a Point in a Line

Point is the **reflection** of point in line if and only if line is the **perpendicular bisector** of .

- The mirror line is a perpendicular ____________________________ of the line that connects the
*object*to its reflected image.

Reflections in Special Lines

In a coordinate plane there are some “special” lines for which it is relatively easy to create **reflections:**

- the axis
- the axis
- the line (this line makes a angle between the axis and the axis)
- The _________-axis, the _________-axis, and the line _________ = _________ are “special” lines to use as
*mirrors*when finding**reflections**of figures.

We can develop simple formulas for reflections in these lines.

Let be a point in the coordinate plane:

We now have the following reflections of

- Reflection of in the axis is

[ the coordinate stays the *same,* and the coordinate is *opposite*]

- Reflection of in the axis is

[ the coordinate is *opposite,* and the coordinate stays the *same* ]

- Reflection of in the line is

[ switch the coordinate and the coordinate ]

Look at the graph above and you will be convinced of the first two **reflections** in the axes. We will prove the third **reflection** in the line on the next page.

- Reflections in the axis have the same ________-coordinate, but the coordinate has the _________________________ value.
- Reflections in the axis have an ___________________________ coordinate, and the coordinate stays the ____________________.
- For reflections in the line, __________________ the and coordinates.

**Example 2**

*Prove that the reflection of point in the line is the point .*

Here is an “outline” proof:

First, we know the **slope** of the line is 1 because .

Next, we will investigate the **slope** of the line that connects our two points, . Use the slope formula and the values of the points’ coordinates given above:

**Slope** of is

Therefore, we have just shown that and are **perpendicular** because the *product* of their **slopes** is –1.

Finally, we can show that is the **perpendicular bisector** of by finding the **midpoint** of :

**Midpoint** of is

We know the **midpoint** of is on the line because the coordinate and the coordinate of the **midpoint** are the same.

Therefore, the line is the **perpendicular bisector** of .

Conclusion: The points and are **reflections** in the line .

**Example 3**

*Point is reflected in the line . The image is . is then reflected in the axis. The image is . What are the coordinates of ?*

We find one reflection at a time:

- Reflect in the line to find :

For reflections in the line we ___________________ coordinates.

Therefore, is (2, 5).

- Reflect in the axis:

For reflections in the axis, the coordinate is _____________________ and the coordinate stays the _______________________.

Therefore, is (–2, 5).

## Reflections Are Isometries

Like a **translation,** a **reflection** in a line is also an **isometry.** Distance between points is “preserved” (stays the same).

- A reflection in a line is an __________________________, which means that
*distance*is*preserved.*

We will verify the **isometry** for **reflection** in the axis. The proof is very similar for **reflection** in the axis.

The diagram below shows and its reflection in the axis, :

Use the Distance Formula:

So

Conclusion: When a segment is **reflected** in the axis, the image segment has the *same length* as the *original* **preimage** segment. This is the meaning of **isometry.** You can see that a similar argument would apply to **reflection** in *any* line.

**Reading Check:**

1. *True or false:* Both translations and reflections are isometries.

2. *What is the meaning of the statement in #1 above?*

3. *If a translation rule is , in which directions is a point moved?*

4. *When a point or figure is reflected in a line, that line acts as a mirror.*

a. *How does the axis change a point that is reflected? What do you do to the coordinates of the point in this type of reflection?*

b. *How does the axis change a point that is reflected? What do you do to the coordinates of the point in this type of reflection?*

c. *How does the line change a point that is reflected? What do you do to the coordinates of the point in this type of reflection?*

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Feb 23, 2012## Last Modified:

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