Special triangles

Here are some special triangles you should be familiar with. Whenever you see one, it’s trying to give you a hint. So pay attention and take the hint.

Half-squares (45-45-90)

Draw a square. Draw its diagonal. You’ve just cut the square in half. Each half-square is not only a triangle, but it’s also both a right triangle and an isosceles triangle. So we also call them isosceles right triangles.

• All half-squares are isosceles right triangles.
• All isosceles right triangles are half-squares.

The hypotenuse of a half-square has a length equal to the leg length times root two (√2).

Conversely, the legs each have lengths equal to the hypotenuse length divided by root two (√2).

These are also called 45-45-90 triangles, because those are always the measurements of their angles.

Half-equilaterals (30-60-90)

Also called a 30-60-90 triangle, this is what you get when you cut an equilateral triangle exactly in half with a line straight down the middle. The resulting triangle has a right angle at the bottom. It also has one of the original 60° angles from the equilateral triangle itself. Its third angle is a 30° angle, because it came from the top angle getting cut exactly in half.

So the three angles of this triangle have angles that measure 30°, 60°, and 90°. (Hence the name 30-60-90 triangles.)

The three sides of these triangles always have lengths in the ratio 1:√3:2. As you know, the small angle is across from the small side, etc, which means that the “weird” √3 side is also across from the 60° angle. (Why? Because √3 is between 1 and 2, so the √3 side is the middle-length side. And therefore it’s across from the middle-size angle.)

There are two facts you need to know absolutely cold:

• If a triangle has angles that measure 30°, 60°, and 90°, then its sides are in the ratio 1 : √3 : 2 (also known as x : x√3 : 2x).
• If a triangle has sides in the ratio 1 : √3 : 2 (also known as x : x√3 : 2x), then its angles measure 30°, 60°, and 90°.

Pythagorean triangles

Right triangles follow the Pythagorean Theorem, which tells us that a² + b² = c² .

Usually, this gives us some irrational numbers. For example, if I have a right triangle whose lengths are 2 and 3, then its hypotenuse must measure √13. And if I have a hypotenuse of 4 and one leg whose length is 3, then the other leg must have a length of √7.

But there are some right triangles whose side lengths are all integers. The most well-known example is 3-4-5: 3² + 4² = 5², so a right triangle can have sides of exactly these lengths.

Any triangle having the same shape as a 3-4-5 triangle – in other words any similar triangle – has sides in that same ratio. So if you scale a 3-4-5 triangle up by an integer scale factor, you get another Pythagorean triangle. For example, 6-8-10, 9-12-15, and so on.

There are more Pythagorean triples than just 3-4-5. (Believe it or not, there are infinitely many of them!) The next most common one after 3-4-5 is 5-12-13. Of course, multiples also work: 10-24-26, 15-36-39, and so on.

There are a few things you need to do here.

• Write out and memorize the first five multiples of the 3-4-5 triple:
  • 3-4-5
  • 6-8-10
  • 9-12-15
  • 12-16-20
  • 15-20-25
• Write out and memorize the first three multiples of the 5-12-13 triple.
• Be on the lookout for these numbers. Whenever you see one of these triples – or even when you see two of the three numbers from one of the triples! – you should be on alert. Seeing part of a Pythagorean triple means you have found a hidden hint.

There are two main ways these get used:

1. If a triangle’s side lengths are all part of the same Pythsgorean triple, then the triangle is a right triangle.
2. If a right triangle has two sides that are part of a triple, then the third side completes the triple.

Just one warning about this last point: the long side has to be the hypotenuse. If you match the hypotenuse to the middle or small number, nothing will work out right.

Non-Euclidean triangles

If you draw a triangle on something that isn’t flat (for example, if you draw a triangle on a basketball or balloon), then the triangle is called non-Euclidean. In this case, pretty much none of the rules apply. For example:

• The sides will be curved instead of being line segments
• The angles won’t add up to 180 degrees
• There can be more than one right or obtuse angle

…in other words, it’s a big mess. So… don’t make non-Euclidean triangles just yet, okay?