 Welcome Geometry fans, this time we're going to be talking about 30, 60, 90 triangles. We originally introduced equilateral triangles as also being equilangular. All of the angles in this triangle are the same because all of the sides are the same. In fact we did the math and found out that each angle is exactly 60 degrees. In this second triangle, if we drop an altitude from one point of the triangle down to the opposite side, it forms a right angle with the base. Not only does it form a right angle, but it also splits that side into two equal pieces that we can call x. If one of those sides is the sum of each of those pieces, then one side of this equilateral triangle is length 2x. Now we ask about the angles. First we have a 90 degree angle because of the altitude. But we also have this angle is going to be 60 degrees because it's still part of that equilateral triangle. Which means that the final angle inside this triangle, if you do the math, is going to be 30 degrees. That's how we get a 30-60-90 triangle. Now we take a look at one of these 30-60-90 triangles to find out how we can find the length of the last side of one of these triangles. We know that this length is going to be x and this length here is going to be 2x. Using the Pythagorean theorem, we can find that this side over here is going to be the square root of 2x squared, the entire side, minus x squared. Because 2x is in parentheses, we have to apply the square to each of the 2 and the x, giving us 4x squared. And we still have the minus x squared on the outside. When we combine terms, we get that this is the square root of 3x squared, which we can split into two pieces. Square root of 3 and the square root of x squared, which just equals x. So we can say that this side over here is really x times the square root of 3. This means that any 30-60-90 triangle can be written with the ratio 1 to the square root of 3 to 2. Keep in mind that the square root of 3 is a number that is in between 1 and 2. So there you have it for 30-60-90 triangles. Go check out some of the examples to find out next.