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# 45-45-90 triangles

Introduction to 45-45-90 Triangles. Created by Sal Khan.

Video transcript

Welcome to the presentation
on 45-45-90 triangles. Let me write that down. How come the pen--
oh, there you go. 45-45-90 triangles. Or we could say 45-45-90 right
triangles, but that might be redundant, because we know any
angle that has a 90 degree measure in it is a
right triangle. And as you can imagine, the
45-45-90, these are actually the degrees of the
angles of the triangle. So why are these
triangles special? Well, if you saw the last
presentation I gave you a little theorem that told you
that if two of the base angles of a triangle are equal-- and
it's I guess only a base angle if you draw it like this. You could draw it like this, in
which case it's maybe not so obviously a base angle, but
it would still be true. If these two angles are equal,
then the sides that they don't share-- so this side and this
side in this example, or this side and this side in this
example-- then the two sides are going to be equal. So what's interesting about
a 45-45-90 triangle is that it is a right triangle
that has this property. And how do we know that it's
the only right triangle that has this property? Well, you could imagine a
world where I told you that this is a right triangle. This is 90 degrees, so
this is the hypotenuse. Right, it's the side opposite
the 90 degree angle. And if I were to tell you that
these two angles are equal to each other, what do those
two angles have to be? Well if we call these two
angles x, we know that the angles in a triangle
add up to 180. So we'd say x plus x
plus-- this is 90-- plus 90 is equal to 180. Or 2x plus 90 is equal to 180. Or 2x is equal to 90. Or x is equal to 45 degrees. So the only right triangle in
which the other two angles are equal is a 45-45-90 triangle. So what's interesting about
a 45-45-90 triangle? Well other than what I just
told you-- let me redraw it. I'll redraw it like this. So we already know this is 90
degrees, this is 45 degrees, this is 45 degrees. And based on what I just told
you, we also know that the sides that the 45 degree
angles don't share are equal. So this side is
equal to this side. And if we're viewing it from a
Pythagorean theorem point of view, this tells us that the
two sides that are not the hypotenuse are equal. So this is a hypotenuse. So let's call this side
A and this side B. We know from the Pythagorean
theorem-- let's say the hypotenuse is equal to C-- the
Pythagorean theorem tells us that A squared plus B squared
is equal to C squared. Right? Well we know that A equals
B, because this is a 45-45-90 triangle. So we could substitute
A for B or B for A. But let's just
substitute B for A. So we could say B squared
plus B squared is equal to C squared. Or 2B squared is
equal to C squared. Or B squared is equal
to C squared over 2. Or B is equal to the square
root of C squared over 2. Which is equal to C-- because
we just took the square root of the numerator and the square
root of the denominator-- C over the square root of 2. And actually, even though this
is a presentation on triangles, I'm going to give you a little
bit of actually information on something called
rationalizing denominators. So this is perfectly correct. We just arrived at B-- and we
also know that A equals B-- but that B is equal to C divided
by the square root of 2. It turns out that in most of
mathematics, and I never understood quite exactly why
this was the case, people don't like square root of
2s in the denominator. Or in general they don't
like irrational numbers in the denominator. Irrational numbers are numbers
that have decimal places that never repeat and never end. So the way that they get rid
of irrational numbers in the denominator is that you do
something called rationalizing the denominator. And the way you rationalize
a denominator-- let's take our example right now. If we had C over the square
root of 2, we just multiply both the numerator and
the denominator by the same number, right? Because when you multiply the
numerator and the denominator by the same number, that's just
like multiplying it by 1. The square root of 2 over
the square root of 2 is 1. And as you see, the reason
we're doing this is because square root of 2 times square
root of 2, what's the square root of 2 times
square root of 2? Right, it's 2. Right? We just said, something times
something is 2, well the square root of 2 times square root
of 2, that's going to be 2. And then the numerator is C
times the square root of 2. So notice, C times the square
root of 2 over 2 is the same thing as C over the
square root of 2. And this is important to
realize, because sometimes while you're taking a
standardized test or you're doing a test in class, you
might get an answer that looks like this, has a square root of
2, or maybe even a square root of 3 or whatever, in
the denominator. And you might not see your
answer if it's a multiple choice question. What you ned to do in that case
is rationalize the denominator. So multiply the numerator and
the denominator by square root of 2 and you'll get
square root of 2 over 2. But anyway, back
to the problem. So what did we learn? This is equal to B, right? So turns out that B is equal
to C times the square root of 2 over 2. So let me write that. So we know that A
equals B, right? And that equals the square
root of 2 over 2 times C. Now you might want to memorize
this, though you can always derive it if you use the
Pythagorean theorem and remember that the sides that
aren't the hypotenuse in a 45-45-90 triangle are
equal to each other. But this is very good to know. Because if, say, you're taking
the SAT and you need to solve a problem really fast, and if you
have this memorized and someone gives you the hypotenuse, you
can figure out what are the sides very fast, or i8f someone
gives you one of the sides, you can figure out the
hypotenuse very fast. Let's try that out. I'm going to erase everything. So we learned just now that A
is equal to B is equal to the square root of 2
over 2 times C. So if I were to give you a
right triangle, and I were to tell you that this angle is 90
and this angle is 45, and that this side is, let's
say this side is 8. I want to figure out
what this side is. Well first of all, let's
figure out what side is the hypotenuse. Well the hypotenuse is the side
opposite the right angle. So we're trying to actually
figure out the hypotenuse. Let's call the hypotenuse C. And we also know this is a
45-45-90 triangle, right? Because this angle is 45, so
this one also has to be 45, because 45 plus 90 plus
90 is equal to 180. So this is a 45-45-90 triangle,
and we know one of the sides-- this side could be A or B-- we
know that 8 is equal to the square root of 2
over 2 times C. C is what we're trying
to figure out. So if we multiply both sides of
this equation by 2 times the square root of 2-- I'm just
multiplying it by the inverse of the coefficient on C. Because the square root of 2
cancels out that square root of 2, this 2 cancels
out with this 2. We get 2 times 8, 16 over the
square root of 2 equals C. Which would be correct, but as
I just showed you, people don't like having radicals
in the denominator. So we can just say C is equal
to 16 over the square root of 2 times the square root of 2
over the square root of 2. So this equals 16 square
roots of 2 over 2. Which is the same thing
as 8 square roots of 2. So C in this example is
8 square roots of 2. And we also knows, since this
is a 45-45-90 triangle, that this side is 8. Hope that makes sense. In the next presentation
I'm going to show you a different type of triangle. Actually, I might even start
off with a couple more examples of this, because I feel I
might have rushed it a bit. But anyway, I'll see you soon
in the next presentation.