30-60-90 triangle example problem

So we have this rectangle right over here, and we're told that the length of AB is equal to 1. So that's labeled right over there. AB is equal to 1. And then they tell us that BE and BD trisect angle ABC. So BE and BD trisect angle ABC. So trisect means dividing it into 3 equal angles. So that means that this angle is equal to this angle is equal to that angle. And what they want us to figure out is, what is the perimeter of triangle BED? So it's kind of this middle triangle in the rectangle right over here. So at first this seems like a pretty hard problem, because you're like well, what is the width of this rectangle. How can I even start on this? They've only given us one side here. But they've actually given us a lot of information, given that we do know this is a rectangle. We have four sides, and that we have four angles. The sides are all parallel to each other and that the angles are all 90 degrees. Which is more than enough information to know that this is definitely a rectangle. And so one thing we do know is that opposite sides of a rectangle are the same length. So if this side is 1, then this side right over there is also 1. The other thing we know is that this angle is trisected. Now we know what the measure of this angle is. It was a right angle, it was a 90 degree angle. So if it's divided into three equal parts, that tells us that this angle right over here is 30 degrees, this angle right over here is 30 degrees, and then this angle right over here is 30 degrees. And then we see that we're dealing with a couple of 30-60-90 triangles. This one is 30, 90, so this other side right over here needs to be 60 degrees. This triangle right over here, you have 30, you have 90, so this one has to be 60 degrees. They have to add up to 180, 30-60-90 triangle. And you can also figure out the measures of this triangle, although it's not going to be a right triangle. But knowing what we know about 30-60-90 triangles, if we just have one side of them, we can actually figure out the other sides. So for example, here we have the shortest side. We have the side opposite of the 30 degree side. Now, if the 30 degree side is 1, then the 60 degree side is going to be square root of 3 times that. So this length right over here is going to be square root of 3. And that's pretty useful because we now just figured out the length of the entire base of this rectangle right over there. And we just used our knowledge of 30-60-90 triangles. If that was a little bit mysterious, how I came up with that, I encourage you to watch that video. We know that 30-60-90 triangles, their sides are in the ratio of 1 to square root of 3 to 2. So this is 1, this is a 30 degree side, this is going to be square root of 3 times that. And the hypotenuse right over here is going to be 2 times that. So this length right over here is going to be 2 times this side right over here. So 2 times 1 is just 2. So that's pretty interesting. Let's see if we can do something similar with this side right over here. Here the 1 is not the side opposite the 30 degree side. Here the 1 is the side opposite the 60 degree side. So once again, if we multiply this side times square root of 3, we should get this side right over here. This is the 60, remember this 1, this is the 60 degree side. So this has to be 1 square root of 3 of this side. Let me write this down, 1 over the square root of 3. And the whole reason, the way I was able to get this is, well, whatever this side, if I multiply it by the square root of 3, I should get this side right over here. I should get the 60 degree side, the side opposite the 60 degree angle. Or if I take the 60 degree side, if I divide it by the square root of 3 I should get the shortest side, the 30 degree side. So if I start with the 60 degree side, divide by the square root of 3, I get that right over there. And then the hypotenuse is always going to be twice the side opposite the 30 degree angle. So this is the side opposite the 30 degree angle. The hypotenuse is always twice that. So this is the side opposite the 30 degree angle. The hypotenuse is going to be twice that. It is going to be 2 over the square root of 3. So we're doing pretty good. We have to figure out the perimeter of this inner triangle right over here. We already figured out one length is 2. We figured out another length is 2 square roots of 3. And then all we have to really figure out is, what ED is. And we can do that because we know that AD is going to be the same thing as BC. We know that this entire length, because we're dealing with a rectangle, is the square root of 3. If that entire length is square root of 3, if this AE is 1 over the square root of 3, then this length right over here, ED is going to be square root of 3 minus 1 over the square root of 3. That length minus that length right over there. And how to find the perimeter is pretty straight forward. We just have to add these things up and simplify it. So it's going to be, just let me write this, perimeter of triangle BED is equal to-- This is short for perimeter. I just didn't feel like writing the whole word.-- is equal to 2 over the square root of 3 plus square root of 3 minus 1 over the square root of 3 plus 2. And now this just boils down to simplifying radicals. You could take a calculator out and get some type of decimal approximation for it. Let's see, if we have 2 square root of 3 minus 1 square root of 3, that will leave us with 1 over the square root of 3. 2 over the square of 3 minus 1 over the square root of 3 is 1 over the square root of 3. And then you have the square root of 3 plus 2. And let's see, I can rationalize this. If I multiply the numerator and the denominator by the square root of 3, this gives me the square root of 3 over 3 plus the square root of 3, which I could rewrite that as plus 3 square roots of 3 over 3. Right? I just multiplied this times 3 over 3 plus 2. And so this gives us-- this is the drum roll part now-- so one square root of 3 plus 3 square roots of 3, and all of that over 3, gives us 4 square roots of 3 over 3 plus 2. Or you could put the 2 first. Some people like to write the non-irrational part before the irrational part. But we're done. We figured out the perimeter. We figured out the perimeter of this inner triangle BED, right there.