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# Rigid Transformations

## Identify transformations, translations, reflections and rotations.

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Practice Rigid Transformations
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Transformations in the Plane

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

#### Watch This

https://www.youtube.com/watch?v=F1M1MncPq2c Brightstorm: Transformations in the Plane

#### 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 F\begin{align*}F\end{align*} that stretches each point horizontally by a factor of two. Below, this transformation is applied to triangle S\begin{align*}S\end{align*} to create triangle S\begin{align*}S^\prime\end{align*}S\begin{align*}S^\prime\end{align*} is considered the image of S\begin{align*}S\end{align*} by F\begin{align*}F\end{align*}. You could also say that F\begin{align*}F\end{align*} maps S\begin{align*}S\end{align*} to S\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 S\begin{align*}S\end{align*} correspond to points on S\begin{align*}S^\prime\end{align*}. For example, you could say that “point A\begin{align*}A\end{align*} maps to point A\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 60\begin{align*}60^\circ\end{align*}, its image will also be 60\begin{align*}60^\circ\end{align*}. A transformation that preserves distance and angles is called a 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¯¯¯¯¯\begin{align*}\overline{AC}\end{align*} is 22+42=20=25\begin{align*}\sqrt{2^2+4^2}=\sqrt{20}=2 \sqrt{5}\end{align*}
• The length of AC¯¯¯¯¯¯¯\begin{align*}\overline{A^\prime C^\prime}\end{align*} is 42+42=32=42\begin{align*}\sqrt{4^2+4^2}=\sqrt{32}=4 \sqrt{2}\end{align*}

Example B

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

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

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

3. What makes a transformation a rigid transformation?

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¯¯¯¯¯\begin{align*}\overline{AC}\end{align*} is 32+12=10\begin{align*}\sqrt{3^2+1^2}=\sqrt{10}\end{align*}
• The length of AC¯¯¯¯¯¯¯\begin{align*}\overline{A^\prime C^\prime}\end{align*} is 32+12=10\begin{align*}\sqrt{3^2+1^2}=\sqrt{10}\end{align*}

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

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.

### Vocabulary Language: English

Angle

Angle

A geometric figure formed by two rays that connect at a single point or vertex.
Reflections

Reflections

Reflections are transformations that result in a "mirror image" of a parent function. They are caused by differing signs between parent and child functions.
Transformations

Transformations

Transformations are used to change the graph of a parent function into the graph of a more complex function.
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.