Similar Triangles

Same shape, different size

Take a photo on your phone and pinch to zoom in. The picture gets bigger, but nothing gets squashed or stretched — a face stays a face, a square stays a square. The enlargement is the same shape as the original, just a different size. In geometry we call two shapes like that similar, and it is one of the most useful ideas in all of mathematics: it lets you measure things you could never reach with a ruler.

Two shapes are similar if they have the same shape but not necessarily the same size — one is simply a scaled copy of the other, blown up or shrunk down. We write \triangle ABC \sim \triangle DEF, and the symbol \sim is read "is similar to".

Being a scaled copy means two things hold together:

corresponding angles are equal — scaling never bends a corner, so the angles match exactly;
corresponding sides are in the same ratio — every side is multiplied by the same number, called the scale factor.

\frac{DE}{AB} = \frac{EF}{BC} = \frac{FD}{CA} = k

Here k is the scale factor: double everything and k = 2; halve everything and k = \tfrac12.

How to tell triangles are similar

As with congruence, you don't need to check everything — a few facts force the rest.

Two triangles are similar if any one of these holds:

AA — two angles of one equal two angles of the other. Since the three angles of a triangle always add to 180^\circ, matching two forces the third to match too, so the shapes line up.
SSS (in the same ratio) — all three pairs of corresponding sides share one common ratio k.
SAS — two pairs of sides are in the same ratio and the included angle between them is equal.

Congruence is just the special case where the scale factor is k = 1.

See the scaling

Step through a small triangle and a copy twice as big. The angles stay the same while every side doubles — so the two triangles are similar with scale factor k = 2.

Worked example 1 — find a missing side

Two triangles are similar. The small one has sides 3, 4 and 5. In the big one, the side matching the 3 is 12. How long is the side matching the 5?

Find the scale factor from a pair you know both of: k = \dfrac{12}{3} = 4. The big triangle is four times as big.
Scale the side you want: the side matching the 5 is 5 \times 4 = 20.

\frac{\text{big side}}{\text{small side}} = k \quad\Longrightarrow\quad \text{big side} = k \times \text{small side}

The whole trick is: find k once from a matching pair, then multiply (to go bigger) or divide (to go smaller) every other side by it.

Worked example 2 — the height of a tree

You can't climb a tree with a tape measure — but its shadow can do the job for you. Stand a 1\text{ m} stick upright in the sun. It casts a 2\text{ m} shadow. At the same moment, the tree casts a 10\text{ m} shadow. How tall is the tree?

The sun's rays arrive at the same angle for both, so the stick + its shadow and the tree + its shadow form two similar right-angled triangles. Matching sides are in the same ratio:

\frac{\text{tree height}}{\text{tree shadow}} = \frac{\text{stick height}}{\text{stick shadow}} \quad\Longrightarrow\quad \frac{h}{10} = \frac{1}{2}

So h = 10 \times \tfrac12 = 5\text{ m}. The tree is five metres tall — measured without leaving the ground.

Worked example 3 — matching corners by their angles

Before you write a single ratio you must know which side matches which. The rule: corresponding sides are opposite equal angles. Say \triangle ABC \sim \triangle DEF with \angle A = \angle D, \angle B = \angle E and \angle C = \angle F. Then the sides pair up in that same order:

AB (between A and B) matches DE (between D and E);
BC matches EF;
CA matches FD.

Writing the similarity statement with the letters in matching order (ABC \sim DEF, not ABC \sim EFD) does the bookkeeping for you: the first letters correspond, the second letters correspond, and so on.

Two mistakes trip people up constantly with similar triangles:

Match corresponding sides — the ones opposite equal angles. If you divide a side of the big triangle by the wrong side of the small one, you get a nonsense scale factor and every answer after it is wrong. Line up the equal angles first, then pair the sides that sit across from them. When in doubt, pick a pair where you know both lengths and use that pair to find k.
Similar is not the same as congruent. Similar means "same shape" — the sizes can differ. Congruent means "same shape and same size" — an exact twin. Congruence is just the special similar case where the scale factor is k = 1. Every congruent pair is similar, but most similar pairs are not congruent.

Around 2600 years ago the Greek thinker Thales supposedly stunned the Egyptians by measuring the height of the Great Pyramid without touching it. He waited until the moment when his own shadow was exactly as long as he was tall — the sun at just the right height — and reasoned that at that instant the pyramid's shadow must equal the pyramid's height too. Same similar-triangle trick you just used with the tree, done in the desert sun.