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