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.
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.
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.
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:
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.
There are two main ways these get used:
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.
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:
…in other words, it’s a big mess. So… don’t make non-Euclidean triangles just yet, okay?
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