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Reference
Dilations
Concept
Dilation
A dilation is a transformation that changes the size of a figure while keeping its shape the same. This transformation involves enlarging or reducing the figure by a certain length scale factor k from a fixed point O called the center of dilation. For example, the image of every point on a leaf lies on the ray that starts at the center of the dilation and passes through its preimage.
As shown in the diagram, O is the center of the dilation, k is the scale factor, A is the preimage, and A′ is the image point. By definition, the scale factor k can also be defined as the ratio of a length in the image to the corresponding length in the preimage.
When the scale factor is greater than 1, the dilation is called an enlargement because the image is larger than the preimage. When the scale factor is between 0 and 1, the dilation is called a reduction because the image is smaller than the preimage.
OA′=k⋅OA⇔k=OAOA′
Rule
Properties of Dilations
The following is a list of a few essential properties of dilations.
- The dilation of a line is a line.
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 ∠B′A′C′ be the dilated image of ∠BAC. According to the first property, AC is parallel to A′C′ and AB is parallel to A′B′
Let M be the intersection point of AB and A′C′, and focus on the parallel legs AC and A′C′. According to the Alternate Interior Angles Theorem, angles ∠MAC and ∠AMA′ are congruent.
Similarly, since the other legs are also parallel, angles ∠B′A′M′ and ∠AMA′ are congruent.
Since ∠AMA′ is congruent to both ∠BAC and ∠B′A′C′, 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.
∠BAC≅∠B′A′C′⇓m∠BAC=m∠B′A′C′
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
expand_more 
2
Measure the distance from the center of dilation to the preimage point
expand_more Use a compass to measure the distance from the center of dilation to the preimage point.
3
Copy the measured distance on the ray
expand_more Keep the same opening of the compass and copy the distance on the ray. If the scale factor is n, copy it n−1 times.
OA′=n×OA
The diagram illustrates the construction for scale factor 3, so the distance is copied twice from the preimage point. Dilation by a Scale Factor 1/n
To construct the image when the scale factor is 1/n, the properties of dilation can be used.
1
Draw two rays starting at the center of dilation
expand_more 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 n times
expand_more Use a compass to copy the same distance n times on the second ray, starting at the center of dilation. The diagram illustrates the process when n=3.
3
Construct parallel lines
expand_more 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
expand_more The intersection point of this parallel line with the first ray is the constructed image.
OA′=n1×OA
Dilation by a Scale Factor p/q
The combination of the previous two methods gives the dilation of a point by a scale factor p/q.
1
Dilation by a scale facor 1/q
expand_more First, construct the image using a scale factor 1/q.
OT=q1×OA
On the diagram q=5.
2
Dilation by a scale facor p
expand_more Copy the distance of this new point from the center of dilation p times, starting at the center of dilation.
OA′=p×OTOT=q1×OA⟹OA′=qp×OA
The diagram below illustrates the dilation by scale factor 3/5.
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.
expand_more Depending on the scale factor, choose the appropriate method to construct the image of the vertices.
2
Connect the image vertices.
expand_more Segments are drawn between the image vertices to create the image polygon.
Method
Dilating a Figure in a Coordinate Plane
When dilating a figure in the coordinate plane, the coordinates of the figure's vertices are multiplied by the scale factor. The new coordinates, the image's vertices, can then be graphed in the coordinate plane.
1
Multiply the Coordinates by the Scale Factor
expand_more First, find the positions for the image's vertices by multiplying the coordinates of the vertices A, B, C and D by the scale factor 3. Multiply each of the x- and y-coordinates by 3.
| Preimage | (3x,3y) | Image |
|---|---|---|
| A(1,2) | (3⋅1,3⋅2) | A′(3,6) |
| B(3,1) | (3⋅3,3⋅1) | B′(9,3) |
| C(3,-1) | (3⋅3,3(-1)) | C′(9,-3) |
| D(1,-1) | (3⋅1,3(-1)) | D′(3,-3) |
The images A′, B′, C′ and D′ are the vertices of the image.
2
Plot the Vertices of the Image in the Coordinate Plane
expand_more Now, plot the image's vertices A′(3,6), B′(9,2), C′(9,-3) and D′(3,-3) on the coordinate plane.
3
Draw the Image by Connecting the Vertices
expand_more Finally, the image can be drawn by connecting the image's vertices with segments.
The quadrilateral A′B′C′D′ has the same shape as the preimage ABCD, but the corresponding sides are three times longer. This makes the dilation an enlargement and, since the center of dilation is the origin, the image is further from the origin than the preimage.
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