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Math101learn.math101.caDilations
A dilation scales every point from a fixed centre by a factor $k$. Distances from the centre are multiplied by $|k|$; angle measures and shape are preserved, so the image is similar to the original.
Dilations formalize scale drawings, maps, imaging, and similarity. They reveal why lengths, areas, and volumes follow different powers of a scale factor.
Intuition and core definition
A dilation scales every point from a fixed centre by a factor $k$. Distances from the centre are multiplied by $|k|$; angle measures and shape are preserved, so the image is similar to the original. For the usual school definition $k>0$, points remain on the same ray from the centre.
Notation, language, and conditions
With centre $C$, image point $P\prime$ satisfies $\overrightarrow{CP\prime}=k\overrightarrow{CP}$. About the origin, $(x,y)\mapsto(kx,ky)$. A factor $k>1$ enlarges, $0<k<1$ reduces, $k=1$ is identity; some courses allow $k<0$, placing images on opposite rays.
Why this idea matters
A dilation preserves angle and shape while multiplying every distance from its centre by a common scale factor.
A dependable method
- Identify the centre and scale factor.
- Draw or imagine the ray from the centre through each point.
- Multiply each centre-to-point distance by $|k|$ in the correct direction.
- For origin-centred coordinate dilation, multiply both coordinates by $k$.
- Check proportional side lengths, equal corresponding angles, and fixed centre.
Worked example
Representations and interpretation
Rays through corresponding points meet at the dilation centre. Coordinate multiplication, ruler distances, and a similarity ratio are three representations of the same scaling transformation.
Reasoning about variations
Area scales by $k^2$ and volume by $|k|^3$, not by $k$. Lines not through the centre map to parallel lines, while lines through the centre map to themselves.
Common mistakes
How to check your work
- Compute several image/original length ratios.
- Verify corresponding points and centre are collinear.
- Check one angle or slope relationship is preserved.
Practice
- Dilate $(4,-6)$ about the origin by $k=\frac12$.
- If $k=3$, by what factor does area scale?
- What happens to angle measures under a dilation?
Answers and brief solutions
Show answers
- $(2,-3)$ Multiply each coordinate by $1/2$.
- $9$ Area uses the square of the linear factor.
- They stay equal Dilations preserve shape and therefore angles.
Synthesis and transfer
Resizing a logo about a fixed anchor point maps each vertex along its centre-ray; comparing corresponding distances verifies the common factor and orientation.
With centre $C$, an image point $P'$ must lie on line $CP$ and satisfy directed ratio $CP'/CP=k$. A factor $k=2$ sends every point twice as far on the same ray; $0<k<1$ places it between the centre and original, while negative factors reverse the ray under conventions that allow them. Lengths scale by $|k|$, areas by $k^2$, and angles remain unchanged. Measuring two separate centre-to-image ratios checks whether an apparent resize is truly one dilation rather than a distortion. Parallel image sides arise from the uniform scale, not from an additional translation rule.
Related topics
Teaching and accessibility note
Explore the idea
Vector and matrix transform
Change one quantity at a time and connect what moves to Dilations.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Dilate $(4,-6)$ about the origin by $k=\frac12$.
- Multiply each coordinate by $1/2$.
End of lesson
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