A dilation is a transformation that makes a figure larger or smaller while keeping the same shape. It is one of the main similarity transformations in geometry because it preserves angle measures and changes all lengths by the same scale factor. On the coordinate plane, dilations are easy to visualize by drawing rays from a fixed center of dilation through each point of the figure.
The image points land on those rays at distances controlled by the scale factor.
The center of dilation stays fixed, and every other point moves along a straight line through that center. If the scale factor k is greater than 1, the image is an enlargement farther from the center. If 0 < k < 1, the image is a reduction closer to the center.
When the center is the origin, the coordinate rule is especially simple: (x, y) maps to (kx, ky).
Understanding Geometry: Dilations on the Plane
To construct a dilation carefully, work point by point. Start with the center and one vertex of the original figure. Draw a line through both points.
Measure the distance from the center to that vertex, then multiply that distance by the scale factor. Mark the new point at the resulting distance in the same direction. Repeat for every vertex, then connect the new vertices in the original order.
A useful check is that corresponding sides of the image should be parallel to the original sides. If one image point seems to be off the line from the center, the construction is wrong even if its distance looks reasonable.
A center outside the origin needs a slightly longer coordinate process. First find how far a point lies horizontally and vertically from the center. Multiply both of those offsets by the scale factor.
Then add the new offsets back to the center coordinates. For example, suppose the center has coordinates two, one, a point has coordinates five, three, and the scale factor is two. The point is three units right and two units up from the center.
Its image must be six units right and four units up from the center. The image therefore has coordinates eight, five. This method prevents the common mistake of multiplying a point's coordinates without accounting for a shifted center.
Scale factors can have values beyond the usual positive examples. A negative scale factor places each image point on the opposite side of the center. Its distance from the center is multiplied by the size of the factor.
A scale factor of negative one creates a half turn around the center, so the figure keeps its size but appears across the center. A scale factor of zero sends every point to the center.
The whole figure collapses into one point, so it no longer has the same shape in the usual similarity sense. These cases help show that a dilation controls both distance and direction.
Dilation appears in maps, blueprints, model buildings, photographs, and digital graphics. A map may use a small scale factor to fit a large region on paper. An architect can use a larger scale drawing to show details clearly.
In geometry problems, compare matching lengths before calculating. If one side becomes three times as long, every corresponding side must become three times as long. The perimeter becomes three times as large too.
Area changes more quickly because both length and width are scaled. A figure enlarged by a factor of three has an area nine times as large. Students should keep track of the center, match each vertex correctly, and remember that length units change by one factor while square units change by the factor multiplied by itself.
Key Facts
- A dilation maps each point P to an image point P' on the ray from the center C through P.
- Scale factor: k = CP' / CP.
- If k > 1, the dilation is an enlargement.
- If 0 < k < 1, the dilation is a reduction.
- For center at the origin: (x, y) -> (kx, ky).
- Dilations preserve angle measures and parallel lines, but multiply lengths by k and areas by k^2.
Vocabulary
- Dilation
- A dilation is a transformation that changes the size of a figure by a scale factor while keeping the same shape.
- Center of dilation
- The center of dilation is the fixed point from which distances to all points are scaled.
- Scale factor
- The scale factor is the number that tells how many times farther each image point is from the center compared with the original point.
- Image
- The image is the new figure produced after a transformation is applied to the original figure.
- Similarity
- Similarity means two figures have the same shape, equal corresponding angles, and proportional corresponding side lengths.
Common Mistakes to Avoid
- Multiplying only one coordinate by the scale factor is wrong because a dilation from the origin scales both x and y coordinates.
- Using the wrong center of dilation is wrong because the image points must lie on rays starting at the chosen center, not necessarily the origin.
- Thinking a dilation changes angle measures is wrong because dilations preserve shape and all corresponding angles remain equal.
- Confusing k with k^2 is wrong because side lengths are multiplied by k, while areas are multiplied by k^2.
Practice Questions
- 1 A triangle has vertices A(2, 1), B(4, 1), and C(2, 3). Dilate it by scale factor k = 3 centered at the origin. What are the coordinates of A', B', and C'?
- 2 Point P(8, -4) is dilated from the origin to P'(2, -1). What is the scale factor k, and is the dilation an enlargement or a reduction?
- 3 A quadrilateral is dilated from a center C with scale factor 2. Explain why the image is similar to the original, and describe what happens to its side lengths, angle measures, and area.