MISSION · Geometry
Dilations and Similarity
The fourth transformation, and the only one that changes size
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The idea in one sentence
A dilation pushes every point directly away from — or pulls it towards — one fixed centre, multiplying its distance from that centre by the same scale factor.
What makes it different
Translations, reflections and rotations are rigid: the image is the same size and shape as the original, so the two are congruent. A dilation is the one standard transformation that is not rigid. The image keeps every angle and every proportion but changes size, so the two shapes are similar rather than congruent.
That is the whole relationship between the two topics. Two shapes are similar exactly when some sequence of rigid motions plus one dilation carries one onto the other.
The two things you need
A dilation is specified by a centre and a scale factor \(k\).
Draw a ray from the centre through any point of the shape. The image of that point sits on the same ray, at \(k\) times the distance from the centre.
- \(k > 1\) enlarges, pushing everything outward.
- \(0 < k < 1\) shrinks, pulling everything in.
- \(k = 1\) changes nothing.
- \(k < 0\) sends points through the centre and out the other side, which enlarges and rotates by 180° at once.
Move the centre and the image lands somewhere else, even with the same \(k\). Both inputs matter, and a question that gives only a factor has not finished describing the dilation.
Coordinates, when the centre is the origin
From the origin, a dilation is one multiplication:
\[(x, y) \; \longrightarrow \; (kx, ky)\]So \((2, 5)\) under \(k = 4\) becomes \((8, 20)\). Nothing else to do.
If the centre is some other point \((a, b)\), subtract it, scale, and add it back:
\[(x, y) \; \longrightarrow \; \big(a + k(x - a), \; b + k(y - b)\big)\]This subtract-scale-restore pattern shows up everywhere — it is the same move as completing the square or shifting a graph. Work relative to the interesting point, then put it back.
Lengths, areas and volumes scale differently
This trips up more people than the dilation itself.
\[\text{lengths} \times k \qquad \text{areas} \times k^2 \qquad \text{volumes} \times k^3\]Under \(k = 3\), every side triples but the area becomes nine times larger, because area is two lengths multiplied. A photo enlarged to triple the width needs nine times the ink.
It is also why scale models are misleading about weight, and why a doubled-in-height animal would need eight times the muscle to carry four times the bone cross-section — the reason very large land animals have such thick legs.
Where it shows up
- Similar triangles, which are exactly triangles related by a dilation.
- Scale drawings and maps, where the scale is the factor.
- Zooming on a screen: pinch-to-zoom is a dilation centred where your fingers are.
- Pantographs and projectors, which are dilations built out of hardware.
Common mix-ups
- Adding instead of multiplying. A dilation multiplies distances. Adding a fixed amount to every coordinate is a translation.
- Scaling the area by \(k\). Areas scale by \(k^2\).
- Ignoring the centre. The image depends on it, and the centre is the one point that never moves.
- Calling a shrink a “negative” dilation. Shrinking is \(0 < k < 1\). A negative \(k\) flips the shape through the centre.
- Expecting angles to change. They never do. That is what keeps the shapes similar.
Check yourself
What does a scale factor of 3 do to area, where does \((2,5)\) land under a dilation of 4 about the origin, and what does a factor between 0 and 1 do?
Areas scale by \(k^2 = 9\). From the origin you just multiply: \((8, 20)\). And a factor under 1 pulls points towards the centre, so the image is smaller.
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