Similarity

Dilations and the scale factor

A dilation enlarges or shrinks a figure from a fixed point called the center. Every point moves along the line from the center, and its distance from the center is multiplied by the same number k, the scale factor. In the picture at the top of the page, k = 3, so OA′ = 3 · OA.

• If k > 1, the image is larger (an enlargement).
• If 0 < k < 1, the image is smaller (a reduction).
• If k = 1, nothing moves.

A dilation multiplies every length by k, keeps every angle the same, and sends each side to a parallel side (or to a side on the same line).

What similar means, and naming order

Two figures are similar if a dilation followed by rigid motions (slides, turns and flips) moves one exactly onto the other. For two triangles that means:

• all three pairs of matching angles are equal, and
• all three pairs of matching sides are in the same ratio, the scale factor.

We write △ABC ~ △DEF, read “triangle ABC is similar to triangle DEF.” As with congruence, the order of the letters tells you which parts match: A ↔ D, B ↔ E, C ↔ F. In the figure, DE/AB = EF/BC = FD/CA = 10.5/7 = 9/6 = 7.5/5 = 1.5.

Congruent triangles are the special case where the scale factor is 1.

Shortcuts: AA, SSS~ and SAS~

To compare sides, pair shortest with shortest, middle with middle and longest with longest, then divide. For SSS~, the three ratios must all be equal. Notice that AA needs no lengths at all. That is why it was useless for congruence but is the most common way to prove similarity.

The side-splitter theorem and the midsegment

Side-splitter theorem: if a line parallel to one side of a triangle crosses the other two sides, it cuts them in the same ratio. In the figure, DE ∥ BC, so AD : DB = AE : EC, and indeed 4 : 6 = 6 : 9. The converse is also true: if a segment cuts two sides in the same ratio, it is parallel to the third side.

Why it works: the parallel line makes equal corresponding angles, so △ADE ~ △ABC by AA.

Midsegment theorem: the segment joining the midpoints of two sides is parallel to the third side and half as long. This is the side splitter with ratio 1 : 1, where the small triangle is a copy of the big one with scale factor ½.

The altitude to the hypotenuse

Draw the altitude from the right angle C to the hypotenuse AB. It cuts △ABC into two smaller right triangles, and all three triangles are similar: △ACB ~ △ADC ~ △CDB. Each small triangle shares an acute angle with the big one, so AA applies.

Matching sides of △ADC and △CDB give AD/CD = CD/DB, so CD² = AD · DB. The altitude is the geometric mean of the two pieces of the hypotenuse. In the figure, CD² = 4 · 9 = 36, so CD = 6.

Perimeter, area and volume

If two figures are similar with scale factor k:

• every length, including the perimeter, is multiplied by k;
• every area is multiplied by k²;
• for similar solids, every volume is multiplied by k³.

So a model built at scale 1 : 10 has 1/100 of the real surface area and 1/1000 of the real volume. To go backwards, take a square root of an area ratio, or a cube root of a volume ratio, to find k.

Real-world uses: shadows, maps and models

Shadows. At any moment, the sun’s rays reach the ground at the same angle everywhere nearby. An upright object, its shadow and the ray over its top form a right triangle, so all of these triangles are similar by AA. Measure a stick and its shadow, measure the shadow of something tall, and one proportion gives the height.

There is an old story about this. Thales of Miletus is said to have measured the height of the Egyptian pyramids from their shadows. The accounts were written centuries after he lived. Diogenes Laertius quotes Hieronymus, a pupil of Aristotle, as saying Thales waited for the moment when our shadows are as long as we are tall, and Pliny says the same. Plutarch tells it with a stick set up at the end of the pyramid’s shadow, so that the pyramid is to the stick as the one shadow is to the other. MacTutor notes that this version comes close to the idea of similar triangles. Nobody knows exactly what Thales did.

Maps and plans. A map at scale 1 : 50,000 is similar to the ground it shows: 1 cm on the map stands for 50,000 cm, or 500 m. Floor plans and blueprints work the same way.

Scale models. Model cars, trains and buildings keep the real shape with every length multiplied by the same factor. Remember that areas and volumes do not scale by that same factor.

Common mistakes

A short history: Euclid’s Book VI

Euclid’s Elements, written around 300 BCE, treats similar figures in Book VI, built on the theory of ratios from Book V.

• Proposition 2 is the side-splitter theorem and its converse.
• Proposition 4: triangles with equal angles have proportional sides (the basis of AA).
• Proposition 5: triangles with proportional sides have equal angles (SSS~).
• Proposition 6: one equal angle with the sides around it proportional (SAS~).
• Proposition 8: the altitude from the right angle makes two triangles similar to the whole and to each other. A corollary says the altitude is a mean proportional between the two pieces of the base.
• Proposition 19: similar triangles are to one another in the “duplicate ratio” of their matching sides. In modern words, the area ratio is k².

The midsegment theorem as taught today is not one of Euclid’s numbered propositions, though it follows from Proposition 2.

You’ll get four kinds: decide whether two triangles are similar and why, find a missing side, solve a shadow problem, and work with area or volume. Type numbers like 12 or 7.5; fractions like 15/2 work too.

Sources

• Euclid, Elements, Book VI, ed. D. E. Joyce, Clark University: Proposition 2 (side splitter); Proposition 4 (equal angles, proportional sides); Proposition 5 (SSS~); Proposition 6 (SAS~); Proposition 8 (altitude in a right triangle, with its corollary); Proposition 19 (duplicate ratio of areas).
• MacTutor History of Mathematics, University of St Andrews: Thales of Miletus (biography, with the accounts of the pyramid measurement); Euclid of Alexandria (biography).

Credits: all diagrams on this page are original SVG drawings made for Jugami. The pyramid story is retold from the ancient accounts summarized by MacTutor; the shadow figure uses a tree, not a pyramid, so the numbers stay simple.