Dilation - of a polygon
A transformation in which a polygon is enlarged or reduced by a given factor around a given center point.
Adjust the slider on the right to change the scale factor. Drag the center point O.
Dilation is where the polygon grows or shrinks but keeps the same overall shape. It's a little like zooming in or out on a camera.
In the figure above, the polygon is a rectangle ABCD.
As you adjust the slider on the right, the transformed rectangle A'B'C'D gets bigger and smaller, but remains the same shape.
The transformed figure is called the dilated image of the the original.
The amount by which the image grows or shrinks is called the "Scale Factor".
Experiment with the scale factor slider to gets a feel for this idea.
- If the scale factor is say 2, the image is enlarged so its dimensions are twice the original.
- If it is 0.5, the image is reduced, with its dimensions half the original.
- When the scale factor is 1, the image is the exact same size as the original.
Remember: In dilation, multiply the dimensions of the original by the scale factor to get the dimensions of the image.
Center of dilation
In the figure above we have made it easy by placing the center of dilation O in the center of the rectangle, but it can be anywhere.
Drag the point O to the right towards the slider so it is outside the rectangle ABCD. Now adjust the scale factor and see the result.
Original and image are similar
In dilation, the image and the original are
in that they are the same shape but not necessarily the same size. They are not
because that requires them to be the same shape and the same size, which they are not (unless the scale factor happens to be 1.0).
If we pick any point P in the original figure and measure the distance OP from the center of dilation,
then the corresponding point P' in the image lies on the line OP and its distance from O is OP times the scale factor.
In the above figure, click 'reset' and 'show distances'.
Note how the distance OB is 12.
Since the scale factor is 2, the corresponding point in the image B' is twice that distance from O and lying on the same line, so OB' is 2 times 12, or 24.
Note that if the scale factor is less than 1, then the image points are closer
to the center of dilation to create an image smaller than the original.
In the figure above set a scale factor of less than 1 and click 'show distances' to see this in action.
How to construct the dilation of an object
We start with a polygon ABC and a point O defining the center of dilation. We will draw the dilation of ABC with a
scale factor of 2, meaning the image will be twice the size of the original:
Draw a ray from the center point O through one of the vertices. Any one will do. Here we choose C:
Measure the distance from O to C. Let's say this is 3 centimeters. Multiply this by the scale factor (2) to get 6cm.
Measure out from O a distance of 6cm and mark a new point:
Repeat the above two steps for the other vertices, creating a total of three points:
Join the three new points with line segments, forming the dilation of the original triangle. By convention,
the new figure is labelled A'B'C'.
Things to try
In the diagram above - click 'reset'
Move the center of dilation to the right, outside the rectangle ABCD. Adjust the scale factor and observe the result
Using the the instructions above under "How to construct the dilation of an object", repeat the process with the center of dilation outside the triangle.
Repeat the construction with a scale factor of 0.5.
Other transformation topics
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