Similar Triangles and
the Pythagorean Theorem

Before embarking on trigonometry, there are a couple of things you need to know well about geometry, namely the Pythagorean theorem and similar triangles. Both of these are used over and over in trigonometry.

The Pythagorean theorem

Let’s agree again to the standard convention for labeling the parts of a right triangle. Let the right angle be labeled C and the hypotenuse c. Let A and B denote the other two angles, and a and b the sides opposite them, respectively.
The Pythagorean theorem is about right triangles, that is, triangles, one of whose angles is a 90° angle. A right triangle is displayed in the diagram to the right. The right angle be labeled C and the hypotenuse c, while A and B denote the other two angles, and a and b the sides opposite them, respectively, often called the legs of a right triangle.

The Pythagorean theorem states that the square of the hypotenuse is the sum of the squares of the other two sides, that is,

c squared equals a squared plus b squared

This theorem is useful to determine one of the three sides of a right triangle if you know the other two. For instance, if two legs are a = 5, and b = 12, then you can determine the hypotenuse c by squaring the lengths of the two legs (25 and 144), adding the two squares together (169), then taking the square root to get the value of c, namely, 13.

Likewise, if you know the hypotenuse and one leg, then you can determine the other. For instance, if the hypotenuse is c = 41, and one leg is a = 9, then you can determine the other leg b as follows. Square the hypotenuse and the first leg (1681 and 81), subtract the square of the first leg from the square of the hypotenuse (1600), then take the square root to get the value of the other leg b, namely 40.

An explanation of the Pythagorean theorem

Although it isn’t necessary to know why the Pythagorean theorem is true, everyone has some curiosity about it. You can find a very formal proof of it by Euclid, which appears in his Elements, Proposition I.47, and the converse of it in Proposition I.48. The converse says that the only triangles for which c2 = a2 + b2 are right triangles in which c is the hypotenuse. Euclid’s proof is not the easiest to follow, and hundreds of others have been given. Here’s a nice one given by Thabit ibn-Qurra (826–901).

Proof: Start with the right triangle ABC with right angle at C. Draw a square on the hypotenuse AB, and translate the original triangle ABC along this square to get a congruent triangle A'B'C' so that its hypotenuse A'B' is the other side of the square (but the triangle A'B'C' lies inside the square). Draw perpendiculars A'E and B'F from the points A' and B' down to the line BC. Draw a line AG to complete the square ACEG.

Note that ACEG is a square on the leg AC of the original triangle. Also, the square EFB'C' has side B'C' which is equal to BC, so it equals a square on the leg BC. Thus, what we need to show is that the square ABB'A' is equal to the sum of the squares ACEG and EFB'C'.

But that's pretty easy by cutting and pasting. Start with the big square ABB'A'. Translate the triangle A'B'C' back across the square to triangle ABC, and translate the triangle AA'G across the square to the congruent triangle BB'F. Paste the pieces back together, and you see you've filled up the squares ACEG and EFB'C'. Therefore, ABB'A' = ACEG + EFB'C', as required.


Similar triangles

Two triangles ABC and DEF are similar if (1) their corresponding angles are equal, that is, angle A equals angle D, angle B equals angle E, and angle C equals angle F, and (2) their sides are proportional, that is, the ratios of the three corresponding sides are equal:

In fact, as Euclid showed, each of these two conditions implies the others. That is to say, if corresponding angles are equal, then the three ratios are equal (Prop. VI.4), but if the three ratios are equal, then corresponding angles are equal (Prop. VI.5). Thus, it is enough to know either that their corresponding angles are equal or that their sides are proportional in order to conclude that they are similar triangles.

Typically, the smaller of the two similar triangles is part of the larger. For example, in the diagram to the left, triangle AEF is part of the triangle ABC, and they share the angle A. When this happens, the opposite sides, namely BC and EF, are parallel lines.

This situation frequently occurs in trigonometry applications, and for many of those, one of the three angles A, B, or C is a right angle.