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
Note that a dilation is not a rigid transformation, because it does not preserve distance. In the dilation above,
Example A
A shape is dilated by a scale factor of
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
Solution: To dilate the line segment, draw a ray starting at point
Two conjectures you might make are that
Example C
Show that
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.

AB2=42+22→AB2=20→AB=20−−√=25√ 
A′B′2=122+62→A′B′2=180→A′B′=180−−−√=65√
Therefore,
Two line segments are parallel if they have the same slope.
 Slope of
AB¯¯¯¯¯:−12  Slope of
A′B′¯¯¯¯¯¯¯:−12
Therefore,
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
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
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
3. Using your answer to #2, show that
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
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.