0:00:00.000,0:00:00.860


0:00:00.860,0:00:03.250
Let's continue with the[br]30, 60, 90 triangles.

0:00:03.250,0:00:06.480


0:00:06.480,0:00:09.640
So just review what we just[br]learned, or hopefully learned--

0:00:09.640,0:00:15.910
at minimum what we just saw,[br]--is if we have a 30, 60, 90 --

0:00:15.910,0:00:18.380
and once again, remember: this[br]is only applies to 30, 60, 90

0:00:18.380,0:00:26.560
triangles --and if I were to[br]say the hypotenuse is of length

0:00:26.560,0:00:31.320
h, we learned that the side[br]opposite the 30-degree angle,

0:00:31.320,0:00:34.340
and this is the shortest side[br]of the triangle, is going to be

0:00:34.340,0:00:37.270
h over 2, or 1/2 times[br]the hypotenuse.

0:00:37.270,0:00:40.240
And we also learned that the[br]longer side, or the side

0:00:40.240,0:00:42.810
opposite the 60-degree side,[br]is equal to the square

0:00:42.810,0:00:46.840
root of 3 over 2 times h.

0:00:46.840,0:00:50.640
So let's do a problem where[br]we use this information.

0:00:50.640,0:00:56.370
Let's say I had this[br]triangle right here.

0:00:56.370,0:00:58.010
It's a 90-degree triangle;[br]let's say that this

0:00:58.010,0:01:00.690
is 30 degrees.

0:01:00.690,0:01:02.750
And we could also figure out[br]obviously if that's 30, this

0:01:02.750,0:01:07.040
is 90, that this is[br]also 60 degrees.

0:01:07.040,0:01:10.510
And let's say that the[br]hypotenuse is 12.

0:01:10.510,0:01:12.300
The length is 12 and we know[br]that this is the hypotenuse

0:01:12.300,0:01:14.980
because it's opposite[br]the right angle.

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What is the side right here?

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Well, is the side opposite the[br]60-degree angle, or is it

0:01:21.840,0:01:23.910
opposite the 30-degree angle?

0:01:23.910,0:01:26.460
It's the 30-degree angle[br]that opens into it, right?

0:01:26.460,0:01:28.650
I drew this triangle a little[br]bit different on purpose.

0:01:28.650,0:01:32.050
The 30-degree angle opens up[br]into this side, and it's

0:01:32.050,0:01:34.060
also the shortest side.

0:01:34.060,0:01:37.360
We learned that the side[br]opposite the 30-degree angle is

0:01:37.360,0:01:40.680
half the hypotenuse, so the[br]hypotenuse is 12;

0:01:40.680,0:01:42.860
this would be 6.

0:01:42.860,0:01:46.310
And this side, which is[br]opposite the 60-degree side, is

0:01:46.310,0:01:49.730
equal to the square root of 3[br]over 2 times the hypotenuse.

0:01:49.730,0:01:54.690
So it's the square root of 3[br]over 2 times 12, or it's just

0:01:54.690,0:01:58.150
equal to 6 square roots of 3.

0:01:58.150,0:02:01.150
Another interesting thing is,[br]of course, the longer

0:02:01.150,0:02:04.600
non-hypotenuse side is square[br]root of 3 times longer

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than the short side.

0:02:06.270,0:02:07.810
I don't confuse you too much.

0:02:07.810,0:02:08.660
Let's do another one.

0:02:08.660,0:02:15.010


0:02:15.010,0:02:20.800
Let's say this is 30 degrees--[br]it's our right triangle --and I

0:02:20.800,0:02:28.390
were to tell you that this side[br]right here is 5, what is

0:02:28.390,0:02:29.900
the length of this side?

0:02:29.900,0:02:33.970


0:02:33.970,0:02:35.750
Well first of all let's[br]figure out what we have.

0:02:35.750,0:02:37.390
5 is which side?

0:02:37.390,0:02:39.540
So if this is 30 degrees,[br]we know that this is

0:02:39.540,0:02:41.990
going to be 60 degrees.

0:02:41.990,0:02:47.010
So 5 is opposite the 60-degree[br]side, and x is the hypotenuse.

0:02:47.010,0:02:49.840
Since x is opposite the[br]90-degree side, it's also

0:02:49.840,0:02:53.010
the longest side of[br]the right triangle.

0:02:53.010,0:02:57.910
So we know from our formula[br]that 5 is equal to the square

0:02:57.910,0:03:00.940
root of 3 over 2 times the[br]hypotenuse, which in

0:03:00.940,0:03:02.850
this example is x.

0:03:02.850,0:03:04.240
And now we just solve for x.

0:03:04.240,0:03:06.770
We can multiply both[br]sides by the inverse

0:03:06.770,0:03:07.865
of this coefficient.

0:03:07.865,0:03:19.710
So if you just multiply 2 times[br]the square root of 3-- can

0:03:19.710,0:03:25.030
ignore this --we get 10 over[br]the square root of three here.

0:03:25.030,0:03:27.140
And, of course, this 2[br]cancels out with this 2.

0:03:27.140,0:03:28.667
This square root of 3 cancels[br]out this square root

0:03:28.667,0:03:30.970
of 3 is equal to x.

0:03:30.970,0:03:33.510
And now if you watched the last[br]couple of presentations, you

0:03:33.510,0:03:36.690
realize that this could be the[br]right answer, but we have a

0:03:36.690,0:03:39.660
square root of 3 in the[br]denominator, which people don't

0:03:39.660,0:03:42.980
like because it's an irrational[br]number in the denominator.

0:03:42.980,0:03:44.690
And I guess we could[br]have a debate as to

0:03:44.690,0:03:46.010
why that might be bad.

0:03:46.010,0:03:49.870
So let's rationalize[br]this denominator.

0:03:49.870,0:03:55.150
We say x is equal to 10 over[br]the square to 3; to rationalize

0:03:55.150,0:03:57.750
this denominator we can[br]multiply the numerator and the

0:03:57.750,0:03:59.910
denominator by the[br]square root of 3.

0:03:59.910,0:04:02.670
Because as long as we multiply[br]the numerator and the

0:04:02.670,0:04:05.280
denominator by the same thing,[br]it's like multiplying by 1.

0:04:05.280,0:04:09.790
So this is equal to 10 square[br]roots of 3 over square root of

0:04:09.790,0:04:12.996
3 times square of 3;[br]well that's just 3.

0:04:12.996,0:04:16.212
So x equals 10 square[br]roots of 3 over 3.

0:04:16.212,0:04:17.870
That's the hypotenuse.

0:04:17.870,0:04:18.990
I know I confused you.

0:04:18.990,0:04:22.920
And, of course, if this is 10[br]square root of 3 over 3--

0:04:22.920,0:04:26.600
that's the hypotenuse --we know[br]that the 30-degree side-- this

0:04:26.600,0:04:28.820
is 30 degrees --we know the[br]30-degree side is half of

0:04:28.820,0:04:35.430
that, so it's 5 square[br]root of 3 over 3.

0:04:35.430,0:04:38.100
Anyway, I think that might[br]give you a sense of the

0:04:38.100,0:04:40.230
30, 60, 90 triangles.

0:04:40.230,0:04:43.980
I think you might be ready now[br]to try some of the level two

0:04:43.980,0:04:46.080
Pythagorean Theorem problems.

0:04:46.080,0:04:47.600
Have fun.

0:04:47.600,0:04:48.392