When you dilate a line segment, how is the original line segment related to the image?

#### Watch This

https://www.youtube.com/watch?v=El7zOrCDzBs

#### 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.

A **dilation** is an example of a transformation that moves each point along a ray through the point emanating from a fixed center point \begin{align*}P\end{align*}, multiplying the distance from the center point by a common scale factor, \begin{align*}k\end{align*}.

\begin{align*}\Delta ABC\end{align*} below has been dilated about point \begin{align*}P\end{align*} by a scale factor of 2. Notice that \begin{align*}P\end{align*}, \begin{align*}A\end{align*}, and \begin{align*}A^\prime\end{align*} are all collinear. Similarly, \begin{align*}P\end{align*}, \begin{align*}B\end{align*}, and \begin{align*}B^\prime\end{align*} are collinear and \begin{align*}P\end{align*}, \begin{align*}C\end{align*}, and \begin{align*}C^\prime\end{align*} are collinear. \begin{align*}PC=3\end{align*} and \begin{align*}PC^\prime=6\end{align*}. The scale factor of this dilation is 2 because \begin{align*}\frac{PC^\prime}{PC}=\frac{6}{3}=2\end{align*}. If you calculate \begin{align*}PA\end{align*}, \begin{align*}PA^\prime\end{align*}, \begin{align*}PB\end{align*} and \begin{align*}PB^{\prime}\end{align*} you will find that \begin{align*}\frac{PA^\prime}{PA}=\frac{PB^\prime}{PB}=2\end{align*} as well.

Note that a dilation is not a **rigid transformation**, because it does not preserve distance. In the dilation above, \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*} is larger than \begin{align*}\Delta ABC\end{align*}. Dilations do, however, preserve angles. A shape and its image after a dilation will be **similar**, meaning they will be the same shape but not necessarily the same size.

**Example A**

A shape is dilated by a scale factor of \begin{align*}\frac{1}{2}\end{align*}. How does the image relate to the original shape?

**Solution:** If the scale factor is less than 1, the image will be smaller than the original shape.

**Example B**

Dilate the line segment below about point \begin{align*}P\end{align*} by a scale factor of 3. Make at least two conjectures about how \begin{align*}\overline{AB}\end{align*} relates to \begin{align*}\overline{A^\prime B^\prime}\end{align*}.

**Solution:** To dilate the line segment, draw a ray starting at point \begin{align*}P\end{align*} through each end point. Use the grid lines to help you find points on these rays that are three times the distance from point \begin{align*}P\end{align*} as the original endpoints were.

Two conjectures you might make are that \begin{align*}A^\prime B^\prime=3AB\end{align*} or \begin{align*}\overline{A^\prime B^\prime} \ \| \ \overline{AB}\end{align*}

**Example C**

Show that \begin{align*}A^\prime B^\prime=3AB\end{align*} and \begin{align*}\overline{A^\prime B^\prime} \ \| \ \overline{AB}\end{align*} for the dilation in Example B.

**Solution:** To find the lengths of the segments, create right triangles with the segments as their hypotenuses.

Use the Pythagorean Theorem to find the lengths of the hypotenuses.

- \begin{align*}AB^2=4^2+2^2 \rightarrow AB^2=20 \rightarrow AB=\sqrt{20}=2\sqrt{5}\end{align*}
- \begin{align*}{A^\prime B^\prime}^2=12^2+6^2 \rightarrow {A^\prime B^\prime}^2=180 \rightarrow A^\prime B^\prime=\sqrt{180}=6\sqrt{5}\end{align*}

Therefore, \begin{align*}A^\prime B^\prime=3AB\end{align*}.

Two line segments are parallel if they have the same slope.

- Slope of \begin{align*}\overline{AB}: -\frac{1}{2}\end{align*}
- Slope of \begin{align*}\overline{A^\prime B^\prime}: -\frac{1}{2}\end{align*}

Therefore, \begin{align*}\overline{A^\prime B^\prime} \ \| \ \overline{AB}\end{align*}.

**Concept Problem Revisited**

When you dilate a line segment, the original line segment will always be parallel to (or on the same line as) the image. Also, if the length of the original line segment is \begin{align*}L\end{align*} and the scale factor is \begin{align*}k\end{align*}, the length of the image will be \begin{align*}kL\end{align*}.

#### 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.

A ** dilation** is an example of a transformation that moves each point along a ray through the point emanating from a fixed center point \begin{align*}P\end{align*}, multiplying the distance from the center point by a common scale factor, \begin{align*}k\end{align*}.

Informally, two shapes are** similar** if they are the same shape but not necessarily the same size.

#### Guided Practice

1. A shape is dilated by a scale factor of 1. How does the image relate to the original shape?

2. Dilate the line segment below about point \begin{align*}P\end{align*} by a scale factor of \begin{align*}\frac{1}{2}\end{align*}.

3. Using your answer to #2, show that \begin{align*}A^\prime B^\prime=\frac{1}{2}AB\end{align*} and \begin{align*}\overline{A^\prime B^\prime} \ \| \ \overline{AB}\end{align*}.

**Answers:**

1. If the scale factor is 1, the shape does not change size or move at all. The image will be equivalent to the original figure.

2. The distance from \begin{align*}P\end{align*} to \begin{align*}A^\prime\end{align*} should be half the distance from \begin{align*}P\end{align*} to \begin{align*}A\end{align*}. Similarly, the distance from \begin{align*}P\end{align*} to \begin{align*}B^\prime\end{align*} should be half the distance from \begin{align*}P\end{align*} to \begin{align*}B\end{align*}.

Notice that \begin{align*}A^\prime\end{align*} is the midpoint of \begin{align*}\overline{PA}\end{align*} and \begin{align*}B^\prime\end{align*} is the midpoint of \begin{align*}\overline{PB}\end{align*}.

3. Use the Pythagorean Theorem to compare the lengths of the two segments.

- \begin{align*}AB^2=4^2+2^2 \rightarrow AB^2=20 \rightarrow AB=\sqrt{20}=2 \sqrt{5}\end{align*}
- \begin{align*}A^\prime B^{\prime 2}=2^2+1^2 \rightarrow A^\prime B^{\prime 2}=5 \rightarrow A^\prime B^\prime=\sqrt{5}\end{align*}

Therefore, \begin{align*}A^\prime B^\prime=\frac{1}{2}AB\end{align*}.

Two line segments are parallel if they have the same slope.

- Slope of \begin{align*}\overline{AB}: \frac{1}{2}\end{align*}
- Slope of \begin{align*}\overline{A^\prime B^\prime}: \frac{1}{2}\end{align*}

Therefore, \begin{align*}\overline{A^\prime B^\prime} \ \| \ \overline{AB}\end{align*}.

#### Practice

1. Describe how to perform a dilation.

2. Explain why a dilation is not an example of a rigid transformation.

3. True or false: angle measures are preserved in a dilation.

4. A shape is dilated by a scale factor of \begin{align*}\frac{3}{2}\end{align*}. How does the image relate to the original shape?

5. In general, if \begin{align*}k > 1\end{align*} will the image be larger or smaller than the original figure?

6. In general, if \begin{align*}k < 1\end{align*} will the image be larger or smaller than the original figure?

7. Dilate the line segment below about point \begin{align*}P\end{align*} by a scale factor of 2.

8. Using your answer to #7, show that \begin{align*}A^\prime B^\prime=2AB\end{align*}.

9. Using your answer to #7, show that \begin{align*}\overline{A^\prime B^\prime} \ \| \ \overline{AB}\end{align*}.

10. If one of the points of your figure IS the center of dilation, what happens to that point when the dilation occurs?

11. Dilate the line segment below about point \begin{align*}P\end{align*} by a scale factor of \begin{align*}\frac{1}{4}\end{align*}.

12. Using your answer to #11, show that \begin{align*}A^\prime B^\prime=\frac{1}{4}AB\end{align*}.

13. Using your answer to #11, show that \begin{align*}\overline{A^\prime B^\prime} \ \| \ \overline{AB}\end{align*}.

You can perform dilations in *Geogebra* just like you can perform other transformations. Start by creating your figure and the point for your center of dilation. Then, select “Dilate an Object from Point by Factor”, then your figure, and then the center of dilation.

Enter the scale factor into the pop up window and your figure will be dilated.

14. Create a triangle in *Geogebra* and dilate it about the origin by a scale factor of 2.

15. Dilate the same triangle about a different point by a scale factor of 2.

16. Compare and contrast the two images from #13 and #14.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 6.1.