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

#### Guidance

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 **image** **of S by F**. You could also say that

**Each of the points in the image are labeled with a “ ′ ” symbol which is read as “prime”. This helps to show how points on**F maps S to S′ .

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 **rigid transformation.**

**Example A**

Is a horizontal stretch an example of a rigid transformation?

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

- The length of
AC¯¯¯¯¯ is22+42−−−−−−√=20−−√=25√ - The length of
A′C′¯¯¯¯¯¯¯ is42+42−−−−−−√=32−−√=42√

**Example B**

A transformation reflects points in shape

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

**Example C**

A transformation translates points in shape

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

**Concept Problem Revisited**

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.

#### Vocabulary

A ** transformation** is a function that takes points in the plane as inputs and gives other points as outputs.

A ** rigid transformation** preserves distance and angles.

#### Guided Practice

1. A transformation rotates points in shape

2. Use algebra to help prove your answer to #1.

3. What makes a transformation a rigid transformation?

**Answers:**

1. Yes, it is a rigid transformation.

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

- The length of
AC¯¯¯¯¯ is32+12−−−−−−√=10−−√ - The length of
A′C′¯¯¯¯¯¯¯ is32+12−−−−−−√=10−−√

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

#### Practice

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.