You slide a book across your desk. You pour soda from a can into a big glass. How are these actions related to transformations?

### Transformations in a Plane

A **transformation** is a function that takes points in the plane as inputs and gives other points as outputs. You can think of a transformation as a rule that tells you how to create new points.

Suppose you have a transformation \begin{align*}F\end{align*} that stretches each point horizontally by a factor of two. Below, this transformation is applied to triangle \begin{align*}S\end{align*} to create triangle \begin{align*}S^\prime\end{align*}. \begin{align*}S^\prime\end{align*} is considered the **image** **of \begin{align*}S\end{align*} by \begin{align*}F\end{align*}**. You could also say that **\begin{align*}F\end{align*} maps \begin{align*}S\end{align*} to \begin{align*}S^\prime\end{align*}.** Each of the points in the image are labeled with a “ ′ ” symbol which is read as “prime”. This helps to show how points on \begin{align*}S\end{align*} correspond to points on \begin{align*}S^\prime\end{align*}. For example, you could say that “point \begin{align*}A\end{align*} maps to point \begin{align*}A\end{align*} prime”.

Some transformations preserve distance and angles. Preserving distance means that if a line segment is 3 units, its image will also be 3 units. Similarly, preserving angles means if an angle is \begin{align*}60^\circ\end{align*}, its image will also be \begin{align*}60^\circ\end{align*}. A transformation that preserves distance and angles is called a **rigid transformation.**

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#### Solve the following problems

Is a horizontal stretch an example of a rigid transformation?

No. You can prove this using the picture above by showing that distance is not preserved.

- The length of \begin{align*}\overline{AC}\end{align*} is \begin{align*}\sqrt{2^2+4^2}=\sqrt{20}=2 \sqrt{5}\end{align*}
- The length of \begin{align*}\overline{A^\prime C^\prime}\end{align*} is \begin{align*}\sqrt{4^2+4^2}=\sqrt{32}=4 \sqrt{2}\end{align*}

A transformation reflects points in shape \begin{align*}K\end{align*} across \begin{align*}\overleftrightarrow{AB}\end{align*} to create shape \begin{align*}K^\prime\end{align*}. Is this reflection a rigid transformation?

Yes, reflections are rigid transformations. You can verify that the distances between the points are preserved.

A transformation translates points in shape \begin{align*}K\end{align*} along vector \begin{align*}\vec{v}\end{align*} to create shape \begin{align*}K^\prime\end{align*}. Is this translation a rigid transformation?

Yes, translations are rigid transformations. You can verify that the distances between the points are preserved.

**Examples**

**Example 1**

Earlier, you were asked how are the actions related to transformations.

Sliding a book across your desk is a rigid transformation because a book is a rigid object that does not change shape. The distances and angles that make up the book don't change once the book is in a new location. Pouring soda, on the other hand, is not a rigid transformation. Liquid is not a rigid object and it can change shape depending on its surroundings. The overall shape of the soda in the can will be different from the overall shape of the soda in the glass.

#### Example 2

A transformation rotates points in shape \begin{align*}K\end{align*} around point \begin{align*}D\end{align*} to create shape \begin{align*}K^\prime\end{align*}. Does this rotation look like a rigid transformation?

Yes, it is a rigid transformation.

#### Example 3

Use algebra to help prove your answer to #2.

With the Pythagorean Theorem, you can show that corresponding sides are the same length. For example:

- The length of \begin{align*}\overline{AC}\end{align*} is \begin{align*}\sqrt{3^2+1^2}=\sqrt{10}\end{align*}
- The length of \begin{align*}\overline{A^\prime C^\prime}\end{align*} is \begin{align*}\sqrt{3^2+1^2}=\sqrt{10}\end{align*}

#### Example 4

What makes a transformation a rigid transformation?

Rigid transformations preserve distance and angles. All corresponding sides will be the same length and all corresponding angles will be the same measure.

### Review

For 1-7, answer true or false and explain your answer.

1. Translations are rigid transformations.

2. Rotations are rigid transformations.

3. Horizontal stretches are rigid transformations.

4. Rigid transformations preserve location in the plane.

5. Corresponding sides in rigid transformations are the same length.

6. If it's not a rigid transformation, it's not a real transformation.

7. Reflections are rigid transformations.

Use the following image for 8-9.

8. Describe the transformation in your own words. Does it look like a rigid transformation?

9. Prove your answer to #8 by comparing the lengths of two sides.

Use the following image for 10-11.

10. Describe the transformation in your own words. Does it look like a rigid transformation?

11. Prove your answer to #10 by comparing the lengths of two sides.

Use the following image for 12-13.

12. Describe the transformation in your own words. Does it look like a rigid transformation?

13. Prove your answer to #12 by comparing the lengths of two sides.

Use the following image for 14-15.

14. Describe the transformation in your own words. Does it look like a rigid transformation?

15. Prove your answer to #14 by comparing the lengths of two sides.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 2.1.