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Learn how to prove the ratios between the sides of a 30-60-90 triangle. Created by Sal Khan.
Video transcript
What I want to do in this video is discuss a special class of triangles called 30-60-90 triangles. And I think you know why they're called this. The measures of its angles are 30 degrees, 60 degrees, and 90 degrees. And what we're going to prove in this video, and this tends to be a very useful result, at least for a lot of what you see in a geometry class and then later on in trigonometry class, is the ratios between the sides of a 30-60-90 triangle. Remember, the hypotenuse is opposite the 90-degree side. If the hypotenuse has length x, what we're going to prove is that the shortest side, which is opposite the 30-degree side, has length x/2, and that the 60 degree side, or the side that's opposite the 60-degree angle, I should say, is going to be square root of 3 times the shortest side. So square root of 3 times x/2, that's going to be its length. So that's where we're going to prove in this video. And then in other videos, we're just going to apply this. We're going to show that this is actually a pretty useful result. Now, let's start with a triangle that we're very familiar with. So let me draw ourselves an equilateral triangle. So drawing the triangles is always the hard part. This is my best shot at a equilateral triangle. So let's call this ABC. I'm just going to assume that I've constructed an equilateral triangle. So triangle ABC is equilateral. And if it's equilateral, that means all of its sides are equal. And let's say equilateral with sides of length x. So this is going to be x, this is going to be x, and this is going to be x. We also know, based on what we've seen from equilateral triangles before, that the measures of all of these angles are going to be 60 degrees. So this is going to be 60 degrees, this is going to be 60 degrees, and then this is going to be 60 degrees. Now, what I'm going to do is I'm going to drop an altitude from this top point right over here. So I'm going to drop an altitude right down, and by definition, when I'm constructing an altitude, it's going to intersect the base right here at a right angle. So that's going to be a right angle, and then this is going to be a right angle. And it's a pretty straightforward proof to show that not only is this an altitude, not only is it perpendicular to this base, but it's a pretty straightforward proof to show that it bisects the base. And you could pause it, if you like and prove it yourself. But it really comes out of the fact that it's easy to prove that these two triangles are congruent. So let me prove it for you. So let's call this point D right over here. So triangles ABD and BDC, they clearly both share this side. So this side is common to both of them right over here. This angle right over here is congruent to this angle over there. This angle right over here is congruent to this angle over here. And so if these two are congruent to each other, then the third angle has to be congruent to each other. So this angle right over here needs to be congruent to that angle right over there. So these two are congruent. And so you can use actually a variety of our congruence postulates. We could say, side-angle-side congruence. We could use angle-side-angle, any of those to show that triangle ABD is congruent to triangle CBD. And what that does for us, and we could use, as I said, we could use angle-side-angle or side-angle-side, whatever we like to use for this. What that does for us is it tells us that the corresponding sides of these triangles are going to be equal. In particular, AD is going to be equal to CD. These are corresponding sides. So these are going to be equal to each other. And if we know that they're equal to each other, and they add up to x-- remember, this was an equilateral triangle of length x-- we know that this side right over here, is going to be x/2. We know this is going to be x/2. Not only do we know that, but we also knew when we dropped this altitude, we showed that this angle has to be congruent to that angle, and their measures have to add up to 60. So if two things are the same and they add up to 60, this is going to be 30 degrees, and this is going to be 30 degrees. So we've already shown one of the interesting parts of a 30-60-90 triangle, that if the hypotenuse-- notice, and I guess I didn't point this out. By dropping this altitude, I've essentially split this equilateral triangle into two 30-60-90 triangles. And so we've already shown that if the side opposite the 90-degree side is x, that the side opposite the 30-degree side is going to be x/2. That's what we showed right over here. Now we just have to come up with the third side, the side that is opposite the 60-degree side. I'll just use the letters that we already have here. This is BD. And we can just use the Pythagorean theorem right here. BD squared plus this length right over here squared plus x/2 squared is going to be equal to the hypotenuse squared. So we get BD squared plus x/2 squared-- this is just straight out of the Pythagorean theorem.-- plus x/2 squared is going to equal this hypotenuse squared. It's going to equal x squared. And just to be clear, I'm looking at this triangle right here. I'm looking at this triangle right over here on the right, and I'm just applying the Pythagorean theorem. This side squared plus this side squared is going to equal the hypotenuse squared. And let's solve now for BD. You get BD squared plus x squared over 4. x squared over 4 is equal to x squared. You could view this as 4x squared over 4. That's the same thing, obviously, as x squared. If you subtract 1/4 x squared from both sides, or x squared over 4 from both sides, you get BD squared is equal to-- 4x squared over 4 minus x squared over 4 is going to be 3x squared over 4. So it's just going to be 3x squared over 4. Take the principal root of both sides. You get BD is equal to the square root of 3 times x. The principal root of 3 is square root of 3. the principal root of x squared is just x, over the principal root of 4 which is 2. And BD is the side opposite the 60-degree side. So we're done. If this hypotenuse is x, the side opposite the 30-degree side is going to be x/2, and the side opposite the 60-degree side is going to be square root of 3 over 2 times x, or the square root of 3x over 2, depending on how you want to view it.