 Welcome, geometry students, to some examples of 30, 60, 90-dright triangles. And in these right triangles, our sides opposite the 30-degree angle is always going to be x the shortest side. The hypotenuse will be 2x, and the side opposite the 60 will be x root 3. So in this example, I'll call it the first example, x is equal to 5. That means the hypotenuse is 10, because that's the same as 2x. The vertical side opposite the 60 is 5 root 3. So if you can remember x2x and x root 3, solve for x and complete the other sides of the triangle. Now our second example is going to be where we're given the hypotenuse. So I'm going to label this example number 2. And again, we can label the sides x, 2x, and x root 3. So in this case, 2x is 12, so x is equal to 6. And if we go over to the side that's vertical and opposite the 60, it's x root 3, which would make it 6 root 3. So in solving right triangles, these first two examples, when you're given the short side or the long side, it's pretty easy to solve for x and finish off the sides of the triangle. Now for our third triangle, this is going to be the most difficult one. We'll number this one number 3. It's when you're given the side opposite the 60. Okay, and in this case, it's 12. So opposite the 30 is x, the long side is 2x, and this side is x root 3. So you really have to solve the equation x root 3 equals 12 to find x. If we divide both sides by root 3, we can get x is 12 over root 3. And remember, you cannot leave your answer like that. We have a radical in the denominator. So we are going to multiply the top and bottom by the square root of 3. Now I just slid that up to give me a little more space. When you finish solving for x, you're going to have x is equal to 12 root 3 over the square root of 3 times the square root of 3, which is 3, because the square root of 3 times the square root of 3 is the square root of 9. Again, you can cancel these numbers outside the radical. 12 over 3 is 4, so my final answer is 4 root 3. And if x is 4 root 3, that's going to be that side. To find the hypotenuse, which is 2x, I would double that, and you only double the 4. So you would get 2x is 8 root 3, and that would be the hypotenuse side.