Properties of Basic Dilations

Properties of Basic Dilations
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  • 0:04 Dilation as Transformation
  • 0:53 Scale Factors
  • 2:18 Center of Dilation
  • 3:22 Some More Examples
  • 4:32 Lesson Summary
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Lesson Transcript
Instructor: David Karsner
A dilation is the geometric term given to the shrinking or enlarging of a shape while keeping its orientation. It is one of the basic four types of transformations.

Dilation is a Transformation

Do you remember the Nintendo game Super Mario Brothers where the main character would eat a mushroom and then grow in size? The computer's programming is using a dilation to make this happen. A dilation is when a shape has kept its orientation but has been changed in size, either getting smaller or getting larger. This is a form of transformation, which is a repositioning of a shape.

There are four basic transformations: reflection, rotation, translation, and dilation.

  • A reflection is when a shape is reflected across a line.
  • A rotation is when a shape has been rotated.
  • A translation is when a shape has been moved (slid) without changing the orientation of the shape.
  • The fourth transformation (and the topic of this lesson) is dilation.

Scale Factors

When it comes to making something a different size, you must always answer the question: how much do you want to change it? Scale factor is the amount that you want to change a shape. The scale factor is always a measure of multiplication. If the scale factor is a number that is greater than one, then the shape will get larger. If the scale factor is a number that is smaller than one, then the shape will shrink.

If you had a scale factor of three, the lengths of your shape will become three times their original size. A shape that is being dilated with a scale factor of 1/2 will be half the size of the original. This can be thought of as multiplying by a half, or dividing by two. If the scale factor is one, then nothing changes, because multiplying anything by one gets you what you started with.

The scale factor multiplies the distance, and distance is one-dimensional. What happens to the area when a shape has been dilated? Let's take a one-inch square and dilate it by a factor of three. The sides are now three inches a-piece. The area of a three-inch square is nine square inches (3 x 3). The area of the original square was 1 square inch (1 x 1). The length of the side of the square tripled. Area is two-dimensional so we need to use the scale factor twice, squaring it. When finding the volume you will use the scale factor three times, cubing it.

Center of Dilation

Whenever a shape is being dilated, you have to know what point it is being dilated from. This point is called the center of dilation. The center of dilation is assumed to be the origin (0,0) of the x, y coordinate grid, unless otherwise stated.

To determine the dilation, you would start with a point on the original shape. How far is this point from the center of dilation in terms of x and y? That distance will then be multiplied to the scale factor. The new point will be that far away from the center of dilation.

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