Key takeaway: A dilation stretches or squeezes a shape away from or toward a chosen center point, multiplying every distance by one number and keeping the shape intact.
Grade 88.G.A.3Geometry and Measurement
Definition of Dilation
A dilation is a resizing around a pivot point, like zooming a photo. Each point slides along the ray from the center, and its distance to the center gets multiplied by the scale factor \(k\). If \(k > 1\) the picture grows, and if \(0 < k < 1\) it shrinks.
Angles remain the same and side lengths are all multiplied by \(k\), so the image is similar to the original, not congruent unless \(k = 1\). Unlike a slide, flip or turn, this transformation changes size. A projector enlarging a slide and a map zoomed out are everyday examples. If the center is the origin, each coordinate is simply multiplied by \(k\).
Formula and notation
When the pivot is the origin, multiply both coordinates by \(k\):
\[(x, y) \to (kx, \, ky)\]
Lengths are multiplied by \(k\). The image of \(P\) is named \(P^{\prime}\).
Key facts
- A multiplier above 1 enlarges the shape; one between 0 and 1 reduces it.
- The center, each point and that point’s image sit on a single line.
- Every angle keeps its measure, and matching sides remain parallel.
- Area grows by the multiplier squared, not by the multiplier itself.
Dilation example
Triangle \(DEF\) has vertices \(D(1, 1)\), \(E(3, 1)\) and \(F(1, 2)\). Dilate it from the origin with scale factor 3, then compare the lengths of \(DE\) and \(D^{\prime}E^{\prime}\).
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Use the rule \((x, y) \to (3x, 3y)\).
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\(D(1, 1) \to D^{\prime}(3, 3)\), \(E(3, 1) \to E^{\prime}(9, 3)\), \(F(1, 2) \to F^{\prime}(3, 6)\).
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\(DE = 3 – 1 = 2\) and \(D^{\prime}E^{\prime} = 9 – 3 = 6\), and \(6 = 3 \times 2\).
The image is \(D^{\prime}(3, 3)\), \(E^{\prime}(9, 3)\), \(F^{\prime}(3, 6)\), and every side is 3 times as long.
Common mistake
Watch out
Sasha dilates \(D(1, 1)\) with scale factor 3 by adding: \(D^{\prime}(1 + 3, 1 + 3) = (4, 4)\).
Why it is wrong
Adding 3 is a slide, which would not resize anything. A dilation multiplies each coordinate by the factor: \(D^{\prime}(3, 3)\). Test: the distance from the origin must triple, and \((3, 3)\) is three times as far from the origin as \((1, 1)\), while \((4, 4)\) is only four times farther along one coordinate.
Related terms and lessons
- Grade 8 · Congruence, Dilations and SimilarityLessonWorksheetQuiz
- Geometry · Dilations and Similarity TransformationsLessonWorksheetQuiz
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