Math101 · Independent Ontario learning library
Math101Dilations
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.
