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Reference

Dilations

Concept

Dilation

A dilation is a point transformation given by a center point and a scale factor .
The image of each point is on the ray starting at the center and going through the preimage. The distance of the image from the center is calculated as follows.

In this formula is the center of the dilation, is the scale factor, is the preimage, and is the image point. This is illustrated on the diagram. Move point and see how the image point moves.
When the scale factor is greater than the dilation is called an enlargement. When the scale factor is between and the dilation is called a reduction.

Rule

Properties of Dilations

The following is a list of a few essential properties of dilations.

  • The dilation of a line is a line.
    • If the line does not pass through the center of dilation, the image line is parallel to the preimage.
    • If the line passes through the center of dilation, the image line and the preimage line are the same.

Proof

The formal proof of this property is beyond the scope of this course. The applet below, however, gives an experimental verification.

  • Move the line, the center of dilation, and set the scale factor.
  • Move a point along the line and verify that the image points indeed form a line parallel to the preimage.


  • The dilation of a line segment is a line segment. The image is longer or shorter than the preimage in the ratio given by the scale factor.

Proof

The formal proof of this property is beyond the scope of this course. The applet below, however, gives an experimental verification.

  • Move the segment, the center of dilation, and set the scale factor.
  • Move a point along the segment and verify that the image points indeed form a segment parallel to the preimage.
  • Check also that the ratio of the length of the image and the length of the preimage is indeed the scale factor. Note that the displayed lengths are only approximate, so the ratio will only be approximately equal to the scale factor.


  • Dilations preserve angle measures.

Proof

Let be the dilated image of According to the first property, is parallel to and is parallel to

Let be the intersection point of and and focus on the parallel legs and According to the Alternate Interior Angles Theorem, angles and are congruent.

Similarly, since the other legs are also parallel, angles and are congruent.

Since is congruent to both and the transitive property of congruence implies that these two angles are congruent.

By the definition of congruence, this completes the proof that dilation preserves angle measures.

Construction

Dilation Using Straightedge and Compass

The methods to construct the image of a point depend on the scale factor.

Dilation by an Integer Scale Factor

When the scale factor is an integer, a compass can be used to copy the distance between the center of dilation and the preimage point to find the position of the image.
1
Draw a ray from the center of dilation through the preimage point
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Use a straightedge to draw a ray from the center of dilation through the preimage point.


2
Measure the distance from the center of dilation to the preimage point
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Use a compass to measure the distance from the center of dilation to the preimage point.

3
Copy the measured distance on the ray
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Keep the same opening of the compass and copy the distance on the ray. If the scale factor is copy it times.
The diagram illustrates the construction for scale factor so the distance is copied twice from the preimage point.

Dilation by a Scale Factor

To construct the image when the scale factor is the properties of dilation can be used.

1
Draw two rays starting at the center of dilation
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Use a straightedge to connect the center of dilation with the preimage point and draw a second ray starting at the center of dilation.


2
Copy the same distance on the second ray times
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Use a compass to copy the same distance times on the second ray, starting at the center of dilation. The diagram illustrates the process when


3
Construct parallel lines
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Mark the first and last point on the second ray. Connect the last point with the preimage point and construct a parallel line through the first point.


4
Mark the image point
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The intersection point of this parallel line with the first ray is the constructed image.


Dilation by a Scale Factor

The combination of the previous two methods gives the dilation of a point by a scale factor

1
Dilation by a scale facor
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First, construct the image using a scale factor
On the diagram


2
Dilation by a scale facor
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Copy the distance of this new point from the center of dilation times, starting at the center of dilation.
The diagram below illustrates the dilation by scale factor .


Dilation of a Figure

The dilation of a figure is the collection of all dilated points. The dilation of a polygon can be constructed by dilating the vertices and connecting the image points.
1
Construct the image of the vertices.
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Depending on the scale factor, choose the appropriate method to construct the image of the vertices.

2
Connect the image vertices.
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Segments are drawn between the image vertices to create the image polygon.