A ratio is a comparison, not a measurement
You already know how to measure a single segment with the ruler postulate: lay it down, read off a length like |AB| = 6. A ratio does something quieter and more powerful — it sets two such lengths side by side and asks only how they compare. If |AB| = 6 and |CD| = 9, the ratio of AB to CD is 6 to 9, written 6/9 or, in lowest terms, 2/3. The actual centimetres have evaporated; what survives is the relationship, and that relationship is the whole subject of this rung.
Three quiet rules keep a ratio honest. First, the two quantities must be measured in the same unit — comparing 6 cm to 9 cm gives 2/3, but 6 cm to 9 metres does not. Second, order matters: the ratio of AB to CD is 6/9, while CD to AB is 9/6, and they are not the same. Third, a ratio is just a number, so it carries no units of its own; the centimetres in the top and the centimetres in the bottom cancel and leave a pure 2/3. That unit-free quality is exactly why a ratio can compare a tabletop to a roof.
A proportion is two ratios that agree
Set two ratios equal and you have a proportion: a/b = c/d. Read it aloud as 'a is to b as c is to d'. The whole engine of similar figures will turn out to be a pile of proportions, so it pays to know one tool cold — cross-multiplication. From a/b = c/d, multiplying both sides by b and by d clears the fractions and gives a*d = b*c. Nothing mystical is happening; you are just undoing two divisions at once.
The four numbers have old names worth knowing, because the means and extremes language shows up in every textbook. Written a/b = c/d, the outer two, a and d, are the extremes; the inner two, b and c, are the means. Cross-multiplication then reads as a single memorable sentence: the product of the extremes equals the product of the means, a*d = b*c. That one line is the test for whether a proportion holds at all.
- You are told 4/6 = 6/x and want the missing length x. Identify the means and extremes: extremes 4 and x, means 6 and 6.
- Set the product of the extremes equal to the product of the means: 4 * x = 6 * 6, so 4x = 36.
- Divide both sides by 4 to isolate x: x = 9. The proportion 4/6 = 6/9 indeed holds, since both ratios reduce to 2/3.
Bending a proportion without breaking it
A proportion is sturdier than it looks: you can rearrange it in several ways and it stays true, which is enormously handy when a problem hands you the ratio upside down or sideways from what you want. These rearrangements are the properties of proportions, and every one of them is just cross-multiplication wearing a different coat. Knowing they exist saves you from re-deriving the same equation four times.
Start: a/b = c/d (means b,c ; extremes a,d) Invert: b/a = d/c flip both sides Swap means: a/c = b/d d/c -> trade the inner terms Add one: (a+b)/b = (c+d)/d add 1 to each side All four say the same thing as a*d = b*c.
The last line, the 'add one' move, deserves a second glance because it is the secret behind a theorem you meet two guides from now. Going from a/b = c/d to (a+b)/b = (c+d)/d looks like a trick, but it is honest: add 1 to each side, write each 1 as b/b and d/d, and combine. When this gets attached to a line cutting across a triangle, it becomes the side-splitter theorem — so this small algebra is quietly load-bearing, not decoration.
What 'similar' actually means
Now we can say it precisely. Two figures are similar when one is a faithful scaled copy of the other — same shape, possibly different size. 'Same shape' is not a feeling; it splits into two exact demands. First, corresponding angles are equal. Second, corresponding sides are in proportion, all sharing one common ratio. A photograph and its enlargement are similar; a photograph and a funhouse-mirror stretch are not, because the stretch keeps angles but ruins the side ratios.
That one common ratio has a name: the scale factor, usually written k. If triangle ABC is similar to triangle DEF with scale factor k, then |DE| = k*|AB|, |EF| = k*|BC|, and |DF| = k*|CA| — every side of the copy is k times its partner in the original. When k = 2 the copy is twice as long edge for edge; when k = 1/3 it is a third the size; and when k = 1 the two figures are not merely similar but congruent, the same size as well as the same shape. Congruence, from the previous rung, is simply similarity with k = 1.
Why this is worth the trouble
Similarity is the tool that measures the unmeasurable. A pole 2 m tall casts a 3 m shadow; at the same moment a tree casts a 30 m shadow. The sun's rays strike both at the same angle, so the pole-and-shadow triangle is similar to the tree-and-shadow triangle, and the heights are in the same ratio as the shadows: tree/2 = 30/3, giving a tree 20 m tall — measured without leaving the ground. This is indirect measurement, and it is how surveyors and the ancient Greeks gauged cliffs, rivers, and even the distance to the Moon.
One honest warning before you run with this. Scaling lengths by k does not scale everything by k. Areas scale by k^2 and volumes by k^3 — double a figure's edges and its area quadruples while its volume grows eightfold. That is the square-cube law, and it is why an ant can carry many times its weight while an elephant cannot simply be scaled up from one. We will keep length-scaling and area-scaling firmly apart; conflating them is the single most common similarity mistake.
With ratio, proportion, scale factor, and the precise definition of similar triangles now in hand, you have the vocabulary for the rest of the rung. The very next guide asks a sharp question: do we really have to check all three angles and all three sides every time, or is there a shortcut? The answer — that surprisingly little information forces full similarity — is where this language starts paying real dividends.