Right triangles are pretty special in their own right. But there are two extra-special right triangles. They are 30-60-90 triangles and 45-45-90 triangles, and they appear all the time on the SAT.
In fact, knowing the rules of these two special triangles will open up all sorts of time-saving possibilities for you on the test. Very, very often, instead of having to work out the Pythagorean theorem, you’ll be able to apply the standard side ratios of either of these two types of triangles, cutting out all the time you need to spend calculating.
30-60-90 Triangles
The guy who named 30-60-90 triangles didn’t have much of an imagination. These triangles have angles of 
, 
, and 
. What’s so special about that? This: The side lengths of 30-60-90 triangles always follow a specific pattern. Suppose the short leg, opposite the 30° angle, has length x. Then the hypotenuse has length 2x, and the long leg, opposite the 60° angle, has length x
. The sides of every 30-60-90 triangle will follow this ratio of 1: 
: 2 .
This constant ratio means that if you know the length of just one side in the triangle, you’ll immediately be able to calculate the lengths of all the sides. If, for example, you know that the side opposite the 30º angle is 2 meters long, then by using the 1: 
: 2 ratio, you can work out that the hypotenuse is 4 meters long, and the leg opposite the 60º angle is 2
meters.
And there’s another amazing thing about 30-60-90 triangles. Two of these triangles joined at the side opposite the 60º angle will form an equilateral triangle.
Here’s why you need to pay attention to this extra-special feature of 30-60-90 triangles. If you know the side length of an equilateral triangle, you can figure out the triangle’s height: Divide the side length by two and multiply it by 
. Similarly, if you drop a “perpendicular bisector” (this is the term the SAT uses) from any vertex of an equilateral triangle to the base on the far side, you’ll have cut that triangle into two 30-60-90 triangles.
Knowing how equilateral and 30-60-90 triangles relate is incredibly helpful on triangle, polygon, and even solids questions on the SAT. Quite often, you’ll be able to break down these large shapes into a number of special triangles, and then you can use the side ratios to figure out whatever you need to know.
45-45-90 Triangles
A 45-45-90 triangle is a triangle with two angles of 45° and one right angle. It’s sometimes called an isosceles right triangle, since it’s both isosceles and right. Like the 30-60-90 triangle, the lengths of the sides of a 45-45-90 triangle also follow a specific pattern. If the legs are of length x (the legs will always be equal), then the hypotenuse has length x
:
Know this 1: 1: 
ratio for 45-45-90 triangles. It will save you time and may even save your butt.
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