Pin a center, push every point out by k — the transformation that builds similar figures.
So far the word "similar" has named a relationship between two finished figures — same shape, any size. A dilation does something more active: it manufactures the second figure from the first. Pin one point — the center — and push every other point straight out along its own ray to k times its distance. A light bulb casting a shadow does exactly this, every edge of the cut-out radiating from the one bright source. Enlarge with k > 1, shrink with 0 < k < 1, and — the new trick this lesson adds — flip to the opposite side of the center with a negative k. Whatever you choose, the image comes out similar to the original, which makes the dilation the bridge from this stage into the coordinate transformations ahead.
Hold a paper cut-out between a bare light bulb and a wall. The shadow on the wall is the same shape as the cut-out, only bigger — and if you trace any corner of the shadow back to the bulb, the line runs dead straight through the matching corner of the cut-out. That is the whole picture of a dilation. The bulb is the center O; the cut-out is the original figure; the wall holds the enlarged image.
Made precise: a dilation (also called a scaling or a homothety) is fixed by two things — a center O and a scale factor k. It sends each point P to the point P′ that lies on the ray OP at distance
OP′ = |k| · OP.
Read that slowly. Every point keeps its direction from O — it stays on its own ray — but its distance from O is multiplied by |k|. The center is the one point that never moves: O maps to O itself, because its distance from O is zero, and zero times anything is still zero.
A dilation is named by two ingredients: a center O (where the rays meet) and a scale factor k (how far along each ray the image lands). The rule is P′ on ray OP with OP′ = |k|·OP. The center is the only fixed point.
The power of a dilation is that one number governs the whole plane. For every point P, the ratio OP′/OP = k — the same k everywhere. Nothing is stretched more in one direction than another; the figure scales uniformly. That uniformity is exactly what keeps the shape intact.
Three consequences follow, and they are the reason dilations matter:
Put those together and you get the headline of the whole lesson: the image is always similar to the original, with ratio of similarity |k|. Equal angles, sides in one fixed ratio — that is precisely the definition of similar from §17.3.
Take center O with a point A at distance 3 from O, and dilate by k = 2. Then A′ sits on ray OA at distance 2 · 3 = 6 from O. A side of the figure measuring 5 becomes 10; a corner of 40° stays 40°. Image and original are similar with ratio 2.
You don't need a light bulb — straightedge and a little measuring do the job. To enlarge △ABC by factor k from center O:
If 0 < k < 1 the construction still works, but each image point lands between O and the original vertex, so the image comes out smaller and tucked toward the center. Either way, a quick visual check confirms the build: each image side comes out parallel to the side it copies. A′B′ ∥ AB, B′C′ ∥ BC, C′A′ ∥ CA. If a side of your image is not parallel to its original, you measured a ray wrong.
Until now k has been positive, and every image point landed on the same side of O as its original — the rays pointed outward. A negative scale factor changes that. With k < 0, each point is sent to the opposite side of O. The ray OP is extended backward through O, and the image lands out the far end.
The cleanest case is k = −1. Then OP′ = OP (the image is the same size), but flipped through O — which is exactly a half-turn about O, the point symmetry you met with parallelograms back in Stage 16. The whole figure is rotated 180° around the center. Other negative values both flip and resize: k = −2 sends the image to the opposite side and doubles it.
So the two halves of k split the work cleanly:
| scale factor k | side of O | size vs original | special name |
|---|---|---|---|
| k > 1 | same | larger | enlargement |
| k = 1 | same | identical | does nothing |
| 0 < k < 1 | same | smaller | reduction |
| k = −1 | opposite | identical | half-turn about O |
| k < −1 | opposite | larger | flip + enlarge |
The sign of k decides the side; the size of |k| decides the scale.
A negative k does not make the image "negative" or smaller by itself — |−2| = 2, so k = −2 still doubles every length. The minus sign only flips the image across O. Read the magnitude for size, the sign for side.
We have earned one direction firmly: a dilation always produces a similar figure. Whatever the center and whatever the nonzero k, the image is similar to the original with ratio |k|. That is a one-way ticket from "transformation" to "similar."
Is the converse true — must every pair of similar figures be one single dilation? Not quite. Two figures can be the same shape yet sit in the plane turned or flipped relative to each other. A dilation alone always leaves corresponding sides parallel; if your similar copy has been rotated, its sides are not parallel to the original's, so no single dilation can produce it. To carry one onto the other you may need a dilation plus a rigid motion — a rotation, a reflection, or a translation.
Figures related by a single dilation about one center earn a special name: they are homothetic. Their hallmark is that every pair of corresponding sides is parallel, and all the lines joining corresponding points pass through the one center O.
Similarity = (a dilation) followed by (a rigid motion). A dilation handles the change of size; the rigid motion handles the change of position — sliding, turning, or flipping. A dilation by itself gives a homothetic copy (corresponding sides parallel); add a turn and the sides need no longer be parallel, but the figures are still similar.
A dilation with center O and scale factor k sends each point P to P′ on ray OP with OP′ = |k|·OP — every point keeps its direction from O but multiplies its distance by |k|. The center stays fixed.
A dilation with center O and scale factor k = 3 sends a point that is 4 units from O to a point how many units from O?
OP′ = |k|·OP = 3 · 4 = 12 units from O.
Under a dilation with k = ½, what does a 10 cm segment become?
Every length scales by |k| = ½, so 10 cm → ½ · 10 = 5 cm. (The image is a shrink, between O and the figure.)
A dilation with k = 2 acts on a figure containing a 35° angle. What is the angle in the image?
35°. Dilations preserve angles — only lengths scale, never the corners.
Describe in words what a dilation with k = −1 does to a figure.
It is a half-turn (180° rotation) about O — a point reflection through the center. Same size (|−1| = 1), but every point flips to the opposite side of O.
A figure and its image under a single dilation are drawn. What is always true about each side of the image compared with the matching side of the original?
They are parallel. (Corresponding sides of a homothetic pair are parallel — that is the visual signature of a single dilation.)
True or false: every pair of similar figures can be matched by a single dilation.
False. A dilation always gives a similar image, but two similar figures may also be turned or flipped relative to each other. In general you need a dilation plus a rigid motion (rotation, reflection, or translation).
Six questions to lock it in. Tap the answer you think is right.
This lesson reframes similarity as something you do, not just something you spot. The dilation is the single transformation that creates similar figures by construction, and §17.6 is where Stage 17 connects to the coordinate transformations students meet next (8.G).
The two interactive widgets are built to confront the two stubborn misconceptions head-on. The push every point out by k slider lets students drive k across zero and watch the image jump sides, so a negative factor stops being mysterious; it also keeps angles visibly fixed as lengths grow, countering the belief that magnifying "opens up" the corners. The build it ray by ray stepper makes the construction concrete one ray at a time, and ends on the parallel-side check that lets students self-verify their own constructions.
Students often believe (1) a dilation changes angles — it does not; only lengths scale, angles are preserved; (2) a negative k must shrink the figure — false, since size depends on |k| and the sign only flips the side; and (3) the image must be larger — but 0 < k < 1 shrinks it. Stress: magnitude for size, sign for side.
Common Core: G-SRT.A.1 (verify experimentally the properties of dilations: a line not through the center maps to a parallel line, and lengths scale by the scale factor) · G-SRT.A.2 (define similarity in terms of dilations) · 8.G.A.3 / 8.G.A.4 (describe the effect of dilations on coordinates; understand similarity as a sequence of a dilation and rigid motions).
Next: 17.7 Similarity in Action — turn the whole machine on the world, and measure a tree or a river you can never reach. (Earlier in the stage: 17.5 Similar Polygons and 17.3 Tests for Similar Triangles.)