Similar figures; scale factor
Same shape, any size — and the one number that says how much bigger.
Congruence asked when two shapes are exactly the same. This lesson asks a looser question: when are they the same shape, whatever size either happens to be? Two triangles, ABC and A′B′C′, below, have all six points draggable and independent of each other.
Two triangles, ABC and A'B'C', all six points draggable and independent of each other. A live reading gives the three ratios of corresponding sides, and says whether the triangles are similar — the three ratios agreeing — and if so, what the scale factor between them is. This needs JavaScript switched on.
The three ratios worth tracking are AB:A′B′, BC:B′C′ and CA:C′A′. Two figures are similar when corresponding angles are equal and corresponding sides are all in the same ratio to each other. That one shared ratio is the scale factor between them — how many times bigger (or smaller) one is than the other.
What “same shape” locks down and what it frees
Congruence, several chapters ago, was about having enough information to fix a shape exactly — same size, same angles, no freedom left. Similarity relaxes exactly one part of that: the size. Everything about the shape’s proportions — how the angles compare to each other, how the sides compare to each other — has to stay locked, but the whole figure is free to grow or shrink uniformly.
That is what a constant ratio means, concretely: if AB is twice A′B′, and the triangles are genuinely similar, then BC has to be exactly twice B′C′ as well, and CA exactly twice C′A′ — not roughly, not approximately, exactly, or the two shapes have quietly stopped being the same shape.
Every later fact about similar figures — scaling area, predicting an unmeasured side, the intercept theorem — depends on trusting that exactness rather than treating it as “close enough.” A ratio that’s merely approximate would make none of those later shortcuts safe to use.
Checking all three ratios agree is more than this course will usually need — the next lesson shows that two matching angles alone are already enough to guarantee every ratio matches too, without measuring a single side.
Finding an unknown side from a scale factor
Triangle A′B′C′ is similar to triangle ABC, scale factor 3. AB is 4 units. What is A′B′?
12 units — every side scales by the same factor, so A′B′ is three times AB.
Two similar triangles have corresponding sides 5 and 8. A second pair of corresponding sides is 15 and . What is ?
The scale factor from the first triangle to the second is , so .
A triangle has sides 3, 4 and 5. Another has sides 3, 4 and 6. Are they similar?
No — two of the three ratios agree (, ) but the third does not (). All three ratios have to agree, not just some of them.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.