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# 30-60-90 triangle side ratios proof

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.