S - Dilations Lesson

Recall: Rigid Transformations

You have learned in the past about 3 different transformations:

• Translation (Slide): A transformation that "slides" each point of a figure the same distance in the same direction
• Rotation: A transformation that turns a figure about a fixed point through a given angle and a given direction
• Reflection: A transformation that "flips" a figure over a line of reflection

As you look at each of these transformations think about whether or not each of these transformations changes the shape or size of the figure.

Transformation	Description	Picture
Reflection	In a reflection, a shape is "flipped" over a given line.	A figure is reflected over line j.
Rotation	A rotation rotates or "turns" a shape around a given point.	The figure is rotated around point A.
Translation	A translation will "slide" a shape from one location to another without turning it.	The figure is translated down and to the right.

 

You will notice that after each of these transformations the shape may have changed position or orientation, but has not changed shape or size. Because translations, rotations, and reflections do not change the size or shape of the figure, they are called isometries. Isometries are transformations that produce congruent figures. They preserve shape and size.

 

Dilations

A dilation is a transformation that maintains the original shape of the figure, but stretches or shrinks the size.  All of our dilations will use the origin, (0, 0) as the center of dilation.

Transformation	Description	Picture
Dilation	In a dilation, the figure is increased or decreased in size. All dilations have a center and scale factor.	The figure is dilated with the center j (the red dot) with scale factor 2. For example, if one of the side lengths of the smaller figure is 3 cm, the corresponding side length in the larger figure would be 6 cm.

 

Let's start with this square. Notice that all of the sides have a length of 2 units.

Now, let's dilate the figure by a scale factor of 3. We can notate this dilation by $\left(3x,\:3y\right)$

Now, let's dilate the figure by a scale factor of 1/2. We can notate this dilation by $\left(\frac{1}{2}x,\:\frac{1}{2}y\right)$

Let's try another dilation.

Question: How did the ordered pairs in the dilation above change? 

• Solution: The x's and y's have all been multiplied by 2

Watch this video to practice a few more problems:

Dilations Practice

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