Corresponding sides are in constant ratio

One known pair of sides is enough to predict every other side, exactly.

The last lesson showed two matching angles force every side ratio to agree. This lesson uses that fact the other way round: one known side tells you the rest.

Two triangles, ABC and A′B′C′. A live reading finds the scale factor from AB and A′B′ alone, predicts B′C′ and C′A′ from it, and checks the prediction against their actual lengths.

Two triangles, ABC and A'B'C', all six points draggable and independent of each other. A live reading finds the scale factor from AB and A'B', predicts what B'C' and C'A' should be from that one number alone, and checks the prediction against their actual lengths. This needs JavaScript switched on.

Drag any point. One known ratio predicts both of the others, exactly, every time.

Try to drag your way to a miss — a position where the prediction and the actual length disagree, even slightly.

One ratio, applied to the other two sides

The single number that connects every pair of corresponding sides in two similar figures is their scale factor — the same word the earlier lesson on similarity used, now put to work rather than only measured.

If ABC and A′B′C′ are similar, every one of AB:A′B′, BC:B′C′ and CA:C′A′ is the same ratio — that is what “similar” means. So once one of those three ratios is known, from just one measured pair of sides, the other two are not a fresh measurement away: they are that same ratio, applied to a side already known.

B'C' = BC \times \frac{A'B'}{AB}, \qquad C'A' = CA \times \frac{A'B'}{AB}

Why one measurement is enough

Two figures rarely arrive with every side measured, and they never need to. One known pair of corresponding sides, plus knowing the figures are similar, is enough to reconstruct every length in the other figure — which is also exactly how a map, a scale model, or a blown-up diagram lets you read off a real-world distance from a measured one on paper.

One to watch out for.

This is the tool the next few lessons build on directly: how area scales with the square of the scale factor, and the intercept theorem, both start from exactly this one predictable ratio.

Two similar triangles have AB = 6, A′B′ = 9. BC = 4. What is B′C′?

Scale factor $9 \div 6 = 1.5$ , so $B'C' = 4 \times 1.5 = 6$ .

Two similar rectangles have one pair of sides 5 and 8. A second pair of corresponding sides is 15 and $x$ . What is $x$ ?

Scale factor $8 \div 5 = 1.6$ , so $x = 15 \times 1.6 = 24$ .

A scale drawing has scale factor 1:200. A wall measures 3 cm on the drawing. How long is the real wall, in metres?

$3 \times 200 = 600$ cm, which is 6 metres — the same one-ratio idea, applied to a real measurement rather than another drawn shape.

Do the exercise

This lesson counts as done once you get its exercise right. Nothing to tick off by hand.