Dilations in Math

How to perform dilations

What is a Dilation?

A dilation is a type of transformation that changes the size of the image. The scale factor, sometimes called the scalar factor, measures how much larger or smaller the image is. Below is a picture of each type of dilation (one that gets larger and one that gest smaller)

Example 1

The picture below shows a dilation with a scale factor of 2. This means that the image, A', is twice as large as the pre-image A. Like other transformations, prime notation is used to distinguish the image fromthe pre-image. The image always has a prime after the letter such as A'.

Picture of a dilation in math
Example 2

Dilations can also reduce the size of shape. The picture below demonstrations a dilation of ½ Any time that the scale factor is a fraction, the image will get smaller.

Picture of a dilation in math

Formula for Dilations

It's always easier to understand a concept by looking at specific examples with pictures, so I suggest looking at the dilation examples below first...before you try to internalize the steps listed below and that explain the general formula for dilating a point with coordinates of (2,4) by a scale factor of 1/2.

1) multiply both coordiantes by scale factor (2 *½ ,4 *½)
2) Simplify (1,2)
3) Graph(if required)  

Demonstration of Dilation, Scale factor of 3

paus
animation of dilation by scale of 3

Practice Problems

Promlem 1

Perform a Dilation of 3 on point A (2,1) which you can see in the graph below.

Multiply the coordinates of the original point (2,1), called the image, by 3.

Image's coordinates = (2 *3, 1*3 ) to get the coordinates of the image (6, 3)

Promlem 2

Perform a Dilation of 4 on point A (2,3) which you can see in the picture below.

Multiply the coordinates of the original point (2,3), called the image , by 4.

Image's coordinates = (2 *4, 3*4 ) to get the coordinates of the image (8, 12)

Promlem 3

Perform a Dilation of ½ on point A (2, 4) which you can see in the picture below.

Use the formula for dilations.

1)multiply both coordiantes by scale factor (2 *½ ,4 *½)
2) Simplify (1,2)
3) Graph(if required) see picture below
Promlem 4

Perform a Dilation of 1/3 on point A (3,6) which you can see in the picture below.

Use the formula for dilations.

1) multiply both coordiantes by scale factor (3 *1/3 ,6 *1/3)
2) Simplify (1,2)
3) Graph(if required) see picture below
Promlem 4

What is the image of Triangle ABC graphed below after a dilation of ½?

Multiply each vertex by the scale factor of ½ ! And plot the new coordinates.

Compositions of transformations: dilations

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