0:00:01.500,0:00:03.430
Sorry for starting the[br]presentation with a cough.

0:00:03.430,0:00:06.220
I think I still have a little[br]bit of a bug going around.

0:00:06.220,0:00:10.980
But now I want to continue[br]with the 45-45-90 triangles.

0:00:10.980,0:00:15.190
So in the last presentation we[br]learned that either side of a

0:00:15.190,0:00:19.830
45-45-90 triangle that isn't[br]the hypotenuse is equal to the

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square route of 2 over 2[br]times the hypotenuse.

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Let's do a couple[br]of more problems.

0:00:26.850,0:00:30.680
So if I were to tell you that[br]the hypotenuse of this

0:00:30.680,0:00:33.010
triangle-- once again,[br]this only works for

0:00:33.010,0:00:35.760
45-45-90 triangles.

0:00:35.760,0:00:37.870
And if I just draw one 45[br]you know the other angle's

0:00:37.870,0:00:39.780
got to be 45 as well.

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If I told you that the[br]hypotenuse here is,

0:00:42.960,0:00:44.690
let me say, 10.

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We know this is a hypotenuse[br]because it's opposite

0:00:46.510,0:00:48.340
the right angle.

0:00:48.340,0:00:50.680
And then I would ask you[br]what this side is, x.

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Well we know that x is equal[br]to the square root of 2 over

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2 times the hypotenuse.

0:00:55.490,0:01:01.440
So that's square root[br]of 2 over 2 times 10.

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Or, x is equal to 5[br]square roots of 2.

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Right?

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10 divided by 2.

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So x is equal to 5[br]square roots of 2.

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And we know that this side[br]and this side are equal.

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Right?

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I guess we know this is an[br]isosceles triangle because

0:01:18.490,0:01:20.280
these two angles are the same.

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So we also that this[br]side is 5 over 2.

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And if you're not[br]sure, try it out.

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Let's try the[br]Pythagorean theorem.

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We know from the Pythagorean[br]theorem that 5 root 2 squared,

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plus 5 root 2 squared is equal[br]to the hypotenuse squared,

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where the hypotenuse is 10.

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Is equal to 100.

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Or this is just 25 times 2.

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So that's 50.

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But this is 100 up here.

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Is equal to 100.

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And we know, of course,[br]that this is true.

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So it worked.

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We proved it using the[br]Pythagorean theorem, and

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that's actually how we[br]came up with this formula

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in the first place.

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Maybe you want to go back to[br]one of those presentations

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if you forget how we[br]came up with this.

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I'm actually now going[br]to introduce another

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type of triangle.

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And I'm going to do it the same[br]way, by just posing a problem

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to you and then using the[br]Pythagorean theorem

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to figure it out.

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This is another type[br]of triangle called a

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30-60-90 triangle.

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And if I don't have time[br]for this I will do

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another presentation.

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Let's say I have a[br]right triangle.

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That's not a pretty one,[br]but we use what we have.

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That's a right angle.

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And if I were to tell you that[br]this is a 30 degree angle.

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Well we know that the[br]angles in a triangle

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have to add up to 180.

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So if this is 30, this is 90,[br]and let's say that this is x.

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x plus 30 plus 90 is equal to[br]180, because the angles in

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a triangle add up to 180.

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We know that x is equal to 60.

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Right?

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So this angle is 60.

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And this is why it's called a[br]30-60-90 triangle-- because

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that's the names of the three[br]angles in the triangle.

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And if I were to tell you that[br]the hypotenuse is-- instead of

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calling it c, like we always[br]do, let's call it h-- and I

0:03:27.130,0:03:30.020
want to figure out the other[br]sides, how do we do that?

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Well we can do that[br]using pretty much the

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Pythagorean theorem.

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And here I'm going to[br]do a little trick.

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Let's draw another copy of this[br]triangle, but flip it over

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draw it the other side.

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And this is the same triangle,[br]it's just facing the

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other direction.

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Right?

0:03:48.910,0:03:51.040
If this is 90 degrees[br]we know that these two

0:03:51.040,0:03:53.140
angles are supplementary.

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You might want to review the[br]angles module if you forget

0:03:55.890,0:03:58.980
that two angles that share kind[br]of this common line would

0:03:58.980,0:04:00.000
add up to 180 degrees.

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So this is 90, this[br]will also be 90.

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And you can eyeball it.

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It makes sense.

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And since we flip it, this[br]triangle is the exact

0:04:06.040,0:04:06.890
same triangle as this.

0:04:06.890,0:04:09.130
It's just flipped[br]over the other side.

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We also know that this[br]angle is 30 degrees.

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And we also know that this[br]angle is 60 degrees.

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Right?

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Well if this angle is 30[br]degrees and this angle is 30

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degrees, we also know that this[br]larger angle-- goes all the way

0:04:26.490,0:04:30.230
from here to here--[br]is 60 degrees.

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Right?

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Well if this angle is 60[br]degrees, this top angle is 60

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degrees, and this angle on the[br]right is 60 degrees, then we

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know from the theorem that we[br]learned when we did 45-45-90

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triangles that if these two[br]angles are the same then the

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sides that they don't share[br]have to be the same as well.

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So what are the sides[br]they don't share?

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Well, it's this side[br]and this side.

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So if this side is h[br]then this side is h.

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Right?

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But this angle is[br]also 60 degrees.

0:05:03.680,0:05:07.600
So if we look at this 60[br]degrees and this 60 degrees, we

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know that the sides that they[br]don't share are also equal.

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Well they share this side, so[br]the sides that they don't share

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are this side and this side.

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So this side is h, we also[br]know that this side is h.

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Right?

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So it turns out that if you[br]have 60 degrees, 60 degrees,

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and 60 degrees that all the[br]sides have the same lengths, or

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it's an equilateral triangle.

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And that's something[br]to keep in mind.

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And that makes sense too,[br]because an equilateral triangle

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is symmetric no matter[br]how you look at it.

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So it makes sense that all of[br]the angles would be the same

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and all of the sides would[br]have the same length.

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But, hm.

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When we originally did this[br]problem we only used half of

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this equilateral triangle.

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So we know this whole side[br]right here is of length h.

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But if that whole side is of[br]length h, well then this side

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right here, just the base of[br]our original triangle-- and I'm

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trying to be messy on purpose.

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We tried another color.

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This is going to be[br]half of that side.

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Right?

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Because that's h over 2,[br]and this is also h over 2.

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Right over here.

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So if we go back to our[br]original triangle, and we said

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that this is 30 degrees and[br]that this is the hypotenuse,

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because it's opposite the right[br]angle, we know that the side

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opposite the 30 degree side[br]is 1/2 of the hypotenuse.

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And just a reminder,[br]how did we do that?

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Well we doubled the triangle.

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Turned it into an[br]equilateral triangle.

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Figured out this whole[br]side has to be the same

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as the hypotenuse.

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And this is 1/2 of[br]that whole side.

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So it's 1/2 of the hypotenuse.

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So let's remember that.

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The side opposite the 30 degree[br]side is 1/2 of the hypotenuse.

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Let me redraw that on another[br]page, because I think

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this is getting messy.

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So going back to what[br]I had originally.

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This is a right angle.

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This is the hypotenuse--[br]this side right here.

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If this is 30 degrees, we just[br]derived that the side opposite

0:07:05.080,0:07:09.830
the 30 degrees-- it's like what[br]the angle is opening into--

0:07:09.830,0:07:12.180
that this is equal to[br]1/2 the hypotenuse.

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If this is equal to 1/2[br]the hypotenuse then what

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is this side equal to?

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Well, here we can use the[br]Pythagorean theorem again.

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We know that this side squared[br]plus this side squared-- let's

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call this side A-- is[br]equal to h squared.

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So we have 1/2 h squared plus A[br]squared is equal to h squared.

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This is equal to h squared[br]over 4 plus A squared,

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is equal to h squared.

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Well, we subtract h[br]squared from both sides.

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We get A squared is equal to h[br]squared minus h squared over 4.

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So this equals h squared[br]times 1 minus 1/4.

0:08:07.930,0:08:14.150
This is equal to 3/4 h squared.

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And once going that's[br]equal to A squared.

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I'm running out of space,[br]so I'm going to go all

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the way over here.

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So take the square root of both[br]sides, and we get A is equal

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to-- the square root of 3/4[br]is the same thing as the

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square root of 3 over 2.

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And then the square root[br]of h squared is just h.

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And this A-- remember,[br]this isn't an area.

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This is what decides the[br]length of the side.

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I probably shouldn't[br]have used A.

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But this is equal to the square[br]root of 3 over 2, times h.

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So there.

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We've derived what all the[br]sides relative to the

0:08:56.320,0:08:59.320
hypotenuse are of a[br]30-60-90 triangle.

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So if this is a 60 degree side.

0:09:01.360,0:09:04.750
So if we know the hypotenuse[br]and we know this is a 30-60-90

0:09:04.750,0:09:08.080
triangle, we know the side[br]opposite the 30 degree side

0:09:08.080,0:09:10.500
is 1/2 the hypotenuse.

0:09:10.500,0:09:14.010
And we know the side opposite[br]the 60 degree side is the

0:09:14.010,0:09:18.410
square root of 3 over 2,[br]times the hypotenuse.

0:09:18.410,0:09:22.250
In the next module I'll show[br]you how using this information,

0:09:22.250,0:09:24.120
which you may or may not want[br]to memorize-- it's probably

0:09:24.120,0:09:26.950
good to memorize and practice[br]with, because it'll make you

0:09:26.950,0:09:30.850
very fast on standardized[br]tests-- how we can use this

0:09:30.850,0:09:34.740
information to solve the sides[br]of a 30-60-90 triangle

0:09:34.740,0:09:35.900
very quickly.

0:09:35.900,0:09:37.780
See you in the next[br]presentation.