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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 $$ \frac{1}{2}$$ 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 $$\red { \frac 1 2 }$$.

1) Multiply both coordiantes by scale factor ($$2 \cdot \red { \frac 1 2 } ,4 \cdot \red { \frac 1 2 }$$)
2) Simplify (1, 2)
3) Graph (if required)  

Demonstration of Dilation, Scale factor of 3

play paus
animation of dilation by scale of 3

Practice Problems

Problem 1

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

dilation picture

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).

Problem 2

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

dilation picture

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).

Problem 3

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

dilation picture

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
Problem 4

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

dilation picture

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
Problem 5

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

dilation picture

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

Compositions of transformations: dilations

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