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Welcome to the presentation[br]on 45-45-90 triangles.

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Let me write that down.

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How come the pen--[br]oh, there you go.

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45-45-90 triangles.

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Or we could say 45-45-90 right[br]triangles, but that might be

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redundant, because we know any[br]angle that has a 90 degree

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measure in it is a[br]right triangle.

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And as you can imagine, the[br]45-45-90, these are actually

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the degrees of the[br]angles of the triangle.

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So why are these[br]triangles special?

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Well, if you saw the last[br]presentation I gave you a

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little theorem that told you[br]that if two of the base angles

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of a triangle are equal-- and[br]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[br]which case it's maybe not so

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obviously a base angle, but[br]it would still be true.

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If these two angles are equal,[br]then the sides that they don't

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share-- so this side and this[br]side in this example, or this

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side and this side in this[br]example-- then the two sides

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are going to be equal.

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So what's interesting about[br]a 45-45-90 triangle is that

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it is a right triangle[br]that has this property.

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And how do we know that it's[br]the only right triangle

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that has this property?

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Well, you could imagine a[br]world where I told you that

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this is a right triangle.

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This is 90 degrees, so[br]this is the hypotenuse.

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Right, it's the side opposite[br]the 90 degree angle.

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And if I were to tell you that[br]these two angles are equal to

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each other, what do those[br]two angles have to be?

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Well if we call these two[br]angles x, we know that the

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angles in a triangle[br]add up to 180.

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So we'd say x plus x[br]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[br]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[br]a 45-45-90 triangle?

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Well other than what I just[br]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[br]degrees, this is 45 degrees,

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this is 45 degrees.

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And based on what I just told[br]you, we also know that the

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sides that the 45 degree[br]angles don't share are equal.

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So this side is[br]equal to this side.

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And if we're viewing it from a[br]Pythagorean theorem point of

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view, this tells us that the[br]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[br]A and this side B.

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We know from the Pythagorean[br]theorem-- let's say the

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hypotenuse is equal to C-- the[br]Pythagorean theorem tells us

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that A squared plus B squared[br]is equal to C squared.

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Right?

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Well we know that A equals[br]B, because this is a

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45-45-90 triangle.

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So we could substitute[br]A for B or B for A.

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But let's just[br]substitute B for A.

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So we could say B squared[br]plus B squared is

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equal to C squared.

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Or 2B squared is[br]equal to C squared.

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Or B squared is equal[br]to C squared over 2.

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Or B is equal to the square[br]root of C squared over 2.

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Which is equal to C-- because[br]we just took the square root of

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the numerator and the square[br]root of the denominator-- C

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over the square root of 2.

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And actually, even though this[br]is a presentation on triangles,

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I'm going to give you a little[br]bit of extra information

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on something called[br]rationalizing denominators.

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So this is perfectly correct.

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We just arrived at B-- and we[br]also know that A equals B-- but

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that B is equal to C divided[br]by the square root of 2.

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It turns out that in most of[br]mathematics, and I never

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understood quite exactly why[br]this was the case, people

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don't like square root of[br]2s in the denominator.

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Or in general they don't[br]like irrational numbers

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in the denominator.

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Irrational numbers are numbers[br]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[br]of irrational numbers in the

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denominator is that you do[br]something called rationalizing

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the denominator.

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And the way you rationalize[br]a denominator-- let's take

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our example right now.

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If we had C over the square[br]root of 2, we just multiply

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both the numerator and[br]the denominator by the

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same number, right?

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Because when you multiply the[br]numerator and the denominator

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by the same number, that's just[br]like multiplying it by 1.

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The square root of 2 over[br]the square root of 2 is 1.

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And as you see, the reason[br]we're doing this is because

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square root of 2 times square[br]root of 2, what's the

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square root of 2 times[br]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[br]something is 2, well the square

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root of 2 times square root[br]of 2, that's going to be 2.

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And then the numerator is C[br]times the square root of 2.

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So notice, C times the square[br]root of 2 over 2 is the same

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thing as C over the[br]square root of 2.

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And this is important to[br]realize, because sometimes

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while you're taking a[br]standardized test or you're

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doing a test in class, you[br]might get an answer that looks

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like this, has a square root of[br]2, or maybe even a square root

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of 3 or whatever, in[br]the denominator.

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And you might not see your[br]answer if it's a multiple

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choice question.

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What you ned to do in that case[br]is rationalize the denominator.

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So multiply the numerator and[br]the denominator by square

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root of 2 and you'll get[br]square root of 2 over 2.

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But anyway, back[br]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[br]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[br]equals B, right?

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And that equals the square[br]root of 2 over 2 times C.

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Now you might want to memorize[br]this, though you can always

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derive it if you use the[br]Pythagorean theorem and

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remember that the sides that[br]aren't the hypotenuse in a

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45-45-90 triangle are[br]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[br]the SAT and you need to solve a

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problem really fast, and if you[br]have this memorized and someone

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gives you the hypotenuse, you[br]can figure out what are the

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sides very fast, or if someone[br]gives you one of the sides,

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you can figure out the[br]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[br]is equal to B is equal to the

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square root of 2[br]over 2 times C.

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So if I were to give you a[br]right triangle, and I were to

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tell you that this angle is 90[br]and this angle is 45, and that

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this side is, let's[br]say this side is 8.

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I want to figure out[br]what this side is.

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Well first of all, let's[br]figure out what side

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is the hypotenuse.

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Well the hypotenuse is the side[br]opposite the right angle.

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So we're trying to actually[br]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[br]45-45-90 triangle, right?

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Because this angle is 45, so[br]this one also has to be 45,

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because 45 plus 90 plus[br]90 is equal to 180.

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So this is a 45-45-90 triangle,[br]and we know one of the sides--

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this side could be A or B-- we[br]know that 8 is equal to the

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square root of 2[br]over 2 times C.

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C is what we're trying[br]to figure out.

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So if we multiply both sides of[br]this equation by 2 times the

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square root of 2-- I'm just[br]multiplying it by the inverse

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of the coefficient on C.

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Because the square root of 2[br]cancels out that square root

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of 2, this 2 cancels[br]out with this 2.

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We get 2 times 8, 16 over the[br]square root of 2 equals C.

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Which would be correct, but as[br]I just showed you, people don't

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like having radicals[br]in the denominator.

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So we can just say C is equal[br]to 16 over the square root of

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2 times the square root of 2[br]over the square root of 2.

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So this equals 16 square[br]roots of 2 over 2.

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Which is the same thing[br]as 8 square roots of 2.

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So C in this example is[br]8 square roots of 2.

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And we also knows, since this[br]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[br]I'm going to show you a

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different type of triangle.

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Actually, I might even start[br]off with a couple more examples

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of this, because I feel I[br]might have rushed it a bit.

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But anyway, I'll see you soon[br]in the next presentation.