Algebra II: Mean and Standard Deviation

WEBVTT
Kind: captions
Language: en

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We're on problem 79.

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A small business owner must hire
seasonal workers as the

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need arises.

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The following list shows the
number of employees hired

00:00:09.950 --> 00:00:11.600
monthly for a five-month
period.

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OK, so we hired 4 in the first
month, 13 in the second

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month and so on.

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If the mean of these data--
remember, data is plural.

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I always say this data, but if
they correctly say if the mean

00:00:21.520 --> 00:00:24.210
of these data is approximately
7.

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So the mean is the
average, right?

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So they're saying the average
of these numbers is 7.

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Fair enough.

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If you add all of these up and
divide by 5, you'll get

00:00:31.070 --> 00:00:32.290
approximately 7.

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What is the population standard

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deviation for these data?

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So, with population-- I'll make
a statistical playlist

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one day where I talk about
the difference between a

00:00:43.020 --> 00:00:44.990
population, standard deviation,
and a sample

00:00:44.990 --> 00:00:46.320
standard deviation
and all that.

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But the population standard
deviation, just you have an

00:00:49.880 --> 00:00:54.890
intuition, it's a measure of,
on average, how far are each

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of these numbers
from the mean?

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And it's not exactly the average
of how far they are

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from the mean, but I'll show
you that in a second.

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It's actually the square root of
the average of the squared

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distances of how far each of
these are from the median.

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And that might sound crazy for
a second, but when you

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actually work it all out
the math isn't too bad.

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So what we'll do first is
figure out the variance.

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That's the notation
for the variance.

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And the standard deviation
actually is just the square

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root of this.

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So if you have the variance,
it's very easy to figure out

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the standard deviation.

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But the variance-- and just to
give you the intuition-- it's

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equal to the average
of the squared

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distances from the mean.

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So essentially I say, OK
how far is 4 from 7?

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Oh it's 3 away.

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And I square it.

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I say, OK, the squared
distance is 9.

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The squared distance here,
let me figure it out.

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So let's actually
write them down.

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Let me actually calculate it.

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So I'll do it here.

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4 minus 7 is minus 3.

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If I were to square
that, I'd get 9.

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So the square of the distance
from 4 to 7 is 9.

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So that's a squared distance.

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So the squared distance
from 13 to 7.

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See, they're 6 apart.

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You square them.

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And because we're squaring it
doesn't matter if we do 7

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minus 12 or 13 minus 7.

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Because a negative times a
negative is a positive.

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Anyway 13 minus 7 is 6.

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That squared is 36.

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Then I have 5 minus 7.

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That's minus 2 squared
is equal to 4.

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Then I have 6 minus 7.

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That's negative 1.

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Squared is equal to 1.

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And then I have 9 minus 7.

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That's 2.

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Squared is equal to 4.

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So the 9 is the square of
how far 4 is from 7.

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36 is a square of how
far 13 is from 7.

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So each of these are the squared
distances of each the

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data points from the mean.

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That's where we got
the 7 from, the

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average of these numbers.

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Now, we said the variance
is the average

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of the squared distances.

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So it's just the average
of these five numbers.

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So what's the average of
these five numbers?

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It's going to be 9
plus 36 is 45.

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Plus 4 is 49.

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Plus 1 is 50.

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Plus 1 is 54.

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So it's going to be 54 divided
by, 1, 2, 3, 4, 5.

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Divided by 5.

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So the variance is equal
to the average

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of the squared distances.

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And that's-- what?

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10 and 4/5 is equal to 10.8.

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So that's the variance.

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They don't want the variance.

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They don't want the population
variance.

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They want the population
standard deviation.

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Well that's just the square
of the variance.

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And you can see, since you're
taking the square of the

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square distances, it kind of
becomes close to the average

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of the real distances.

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But it's actually not that
and I don't want to

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confuse you right now.

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For the sake of this problem,
let's just take the square

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root of this.

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So the standard deviation
is just equal to the

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square root of the.

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Variance Which is equal to
the square root of 10.8.

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And what is that?

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Let's see, 3 squared is 9.

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4 squared is 16.

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So the square root of 10.8 is
going to be pretty close to 3.

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Between 3 and 4 but
a lot closer to 3.

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And if we look at all the
choices, the only one that

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even comes close is this one.

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I think if you actually found
the square root it's actually

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very close to 3.3.

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Next problem.

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James found the mean and
standard deviation of the set

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of numbers given above.

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If he adds 5 to each number,
which of the

00:04:35.960 --> 00:04:37.380
following will result?

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All right.

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If you add 5.

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The mean will be multiplied
by 5.

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No, no, no.

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That's not right.

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Because if you add 5 to each
of these numbers, the mean

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will just become 5 bigger.

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You're kind of just
shifting over the

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center to some degree.

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Where you're measuring the
center by the average over 5.

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Actually I could even
show it to you.

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If I said 3 plus 5, and I'm
going to add that to 6 plus 5

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plus 2 plus 5.

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I'm adding 5 to every number.

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Plus 2 plus 5.

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Plus 1 plus 5.

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Plus 7 plus 5.

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Plus 5 plus 5.

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All of that divided
by-- there's, 1,

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2, 3, 4, 5, 6 numbers.

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All of that divided by 6.

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This is the same thing as-- let
me write this-- if we, we

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can get rid of parentheses.

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All we're doing is adding.

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So that's the same thing
as 3 plus 6 plus 2 plus

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1 plus 7 plus 5.

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I just took the first terms
in each of these.

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Plus-- how many 5s do we have?

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1, 2, 3, 4, 5, 6.

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Plus 6 times 5.

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All of that over 6.

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Well, this is going to be equal
to 3 plus 6 plus 2 plus

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1 plus 7 plus 5, over 6.

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Let me scroll down
a little bit.

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Plus-- what's 6 times
5 divided by 6?

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Well that's just 5.

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So this is the average of the
original set of numbers.

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And now, when you add 5 to all
of them, the average of all of

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them combined is just going
to be the original

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average plus 5.

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So you're just shifting the
average to the right.

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So this is not right.

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You're not multiplying
the mean by 5.

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You're just adding
5 to the mean.

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That's not right.

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The standard deviation
will increase by 5.

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Well, in the last video you
saw that the standard

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deviation is the square
root of the variance.

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But both of them are a measure
of the average distance of the

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numbers from the mean,
or from the average.

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So even if you shift everything
by 5-- the mean

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shifts by 5, all the numbers
shift by 5-- their distances

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from the mean aren't going
to change, actually.

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The standard deviation actually
won't change.

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So this isn't right.

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The standard deviation
won't increase by 5.

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These numbers aren't going
to become further

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away from the average.

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So that's not right.

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The mean will not change.

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Well, no.

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We know that the mean
will change.

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I just showed you that
the mean will

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actually increase by 5.

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When all the numbers have
increased by 5.

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That's not right.

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So we already know that D
is probably the answer,

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but let's make sure.

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The standard deviation
will not change.

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Right.

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In general, the average
dispersion, or how far on

00:07:26.040 --> 00:07:28.180
average the numbers are away
from the mean, won't change.

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All the numbers are going to
increase by 5 and the mean is

00:07:30.650 --> 00:07:31.900
going to increase by 5.

00:07:31.900 --> 00:07:35.090
So how far the numbers are from
the mean, their variance

00:07:35.090 --> 00:07:35.880
won't change.

00:07:35.880 --> 00:07:37.250
The square of their distances
won't change.

00:07:37.250 --> 00:07:39.730
And if the variance doesn't
change, then

00:07:39.730 --> 00:07:40.890
the mean won't change.

00:07:40.890 --> 00:07:43.220
So the answer's definitely D.

00:07:43.220 --> 00:07:44.550
And we're all done.

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See you in the next playlist.


Banking 18: Big Picture Discussion

WEBVTT
Kind: captions
Language: en

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In this playlist, we've been
constructing a monetary or

00:00:04.250 --> 00:00:07.850
banking system that reasonably--
in the simplified

00:00:07.850 --> 00:00:10.160
form, approximates the
system we're in.

00:00:10.160 --> 00:00:13.560
So we said, OK, we need
a banking system.

00:00:19.310 --> 00:00:21.340
And we went the fractional
reserve route.

00:00:21.340 --> 00:00:25.810
We said, we'll take deposits,
but we realize that most of

00:00:25.810 --> 00:00:27.010
the deposits just sit there.

00:00:27.010 --> 00:00:31.200
People really don't use them so
let's re-lend out some of--

00:00:31.200 --> 00:00:34.160
90% or 50% of the deposit--
some amount.

00:00:34.160 --> 00:00:37.580
So we have to keep some fraction
aside as reserves and

00:00:37.580 --> 00:00:39.370
that way we can use that money
to lend more, but in the

00:00:39.370 --> 00:00:40.850
process, we create a multiplier

00:00:40.850 --> 00:00:41.860
effect and create money.

00:00:41.860 --> 00:00:44.105
So that's what we described; a
fractional reserve system.

00:00:53.930 --> 00:00:56.190
And when we talk about reserves,
what we originally

00:00:56.190 --> 00:00:58.430
talked about reserves
was gold.

00:00:58.430 --> 00:00:59.750
It was gold based.

00:00:59.750 --> 00:01:01.740
We started off with
the farmers.

00:01:01.740 --> 00:01:02.670
They had their gold.

00:01:02.670 --> 00:01:05.920
Gold was the unit of exchange
from primitive times.

00:01:05.920 --> 00:01:08.210
It was the first agreed
upon medium, and

00:01:08.210 --> 00:01:09.400
people kept with that.

00:01:09.400 --> 00:01:12.360
And then we eventually showed
that in a fractional reserve

00:01:12.360 --> 00:01:14.050
system when you have this
multiplier effect and you

00:01:14.050 --> 00:01:18.350
become kind of an advanced
Western industrial economy, to

00:01:18.350 --> 00:01:20.770
some degree the gold might just
sit in the central bank

00:01:20.770 --> 00:01:23.240
vaults and just kind of
provide this kind of

00:01:23.240 --> 00:01:25.760
confidence in the system, but
really, it does nothing there.

00:01:25.760 --> 00:01:27.180
It just provides a confidence.

00:01:27.180 --> 00:01:29.990
And at some point we show--
I think it was in the last

00:01:29.990 --> 00:01:32.650
video-- that you can actually
just go off the gold standard

00:01:32.650 --> 00:01:36.380
and you could go onto a
non-hard currency-- or

00:01:36.380 --> 00:01:40.520
not-backed by gold.

00:01:40.520 --> 00:01:42.540
And the point of that is really
to get to where we are

00:01:42.540 --> 00:01:45.510
today and to show that
it isn't crazy.

00:01:45.510 --> 00:01:50.070
There's a lot of-- I guess you
could call them conspiracy

00:01:50.070 --> 00:01:53.540
theorists who think that there
was kind of a negative intent

00:01:53.540 --> 00:01:56.020
behind going off of the gold
standard and that it was

00:01:56.020 --> 00:01:56.870
somehow stealing money.

00:01:56.870 --> 00:01:57.600
And you know what?

00:01:57.600 --> 00:02:00.550
I'm not 100% sure that they're
not wrong, but I'm just saying

00:02:00.550 --> 00:02:03.410
since then, it's not like we've
had wealth destruction

00:02:03.410 --> 00:02:07.660
and that we've turned
into some type of

00:02:07.660 --> 00:02:09.699
post-apocalyptic society.

00:02:09.699 --> 00:02:11.520
Economic growth has been pretty
fast. There's been

00:02:11.520 --> 00:02:12.450
innovation.

00:02:12.450 --> 00:02:13.570
Productivity has increased.

00:02:13.570 --> 00:02:16.440
So at least as far as I can
tell, it seems like you can

00:02:16.440 --> 00:02:19.570
have a reasonable economic
system that's not backed by

00:02:19.570 --> 00:02:23.540
gold, that's actually backed
by the actual wealth, where

00:02:23.540 --> 00:02:26.650
the wealth is the productive
capacity of a country.

00:02:26.650 --> 00:02:27.690
So, not backed by gold.

00:02:27.690 --> 00:02:29.900
And that's where we got
to in the last video.

00:02:29.900 --> 00:02:32.340
I think I struck a nerve a
little bit because the gold

00:02:32.340 --> 00:02:35.080
really isn't fundamentally doing
something and it really

00:02:35.080 --> 00:02:37.780
is just a medium of exchange.

00:02:37.780 --> 00:02:40.100
And some people said, how
can you say that?

00:02:40.100 --> 00:02:41.960
Gold is definitely
worth something.

00:02:41.960 --> 00:02:43.740
There's no reality
in which gold

00:02:43.740 --> 00:02:44.830
cannot be worth something.

00:02:44.830 --> 00:02:46.290
Gold controls other assets.

00:02:46.290 --> 00:02:50.530
So it it's good to, I think,
one, think about what other

00:02:50.530 --> 00:02:52.730
routes we could've taken
down this banking

00:02:52.730 --> 00:02:54.580
system evolution tree.

00:02:54.580 --> 00:02:58.270
And then also think about-- I
guess, revisit the notion of

00:02:58.270 --> 00:02:59.870
gold itself.

00:02:59.870 --> 00:03:02.670
So if we just think about--
let's revisit gold first

00:03:02.670 --> 00:03:05.340
because I think that's what a
lot of people care about.

00:03:05.340 --> 00:03:07.910
So there's two things
that I was drawing

00:03:07.910 --> 00:03:08.790
a difference between.

00:03:08.790 --> 00:03:10.800
And it's a philosophical
difference, but I think it's

00:03:10.800 --> 00:03:11.420
an important one.

00:03:11.420 --> 00:03:15.410
It really helps, I guess,
get a handle on

00:03:15.410 --> 00:03:16.600
things in the world.

00:03:16.600 --> 00:03:23.720
So there's money and then
there's wealth-- and this is

00:03:23.720 --> 00:03:26.080
tricky because at the end, these
are just definitions and

00:03:26.080 --> 00:03:29.130
definitions take on different
meanings as time goes on.

00:03:29.130 --> 00:03:32.360
I looked up wealth in Wikipedia
and it gave the Adam

00:03:32.360 --> 00:03:35.550
Smith definition-- from The
Wealth of Nations-- and he

00:03:35.550 --> 00:03:40.030
talks about wealth as the land
and-- I think it was the land

00:03:40.030 --> 00:03:43.340
and labor that produce
utility for humans or

00:03:43.340 --> 00:03:45.050
something of that sort.

00:03:45.050 --> 00:03:47.590
So essentially his definition
is, wealth would be things

00:03:47.590 --> 00:03:53.910
like land, it would be food, it
would be shelter, it'd be

00:03:53.910 --> 00:03:59.160
transportation, it would be,
I don't know, it would be

00:03:59.160 --> 00:04:02.210
entertainment to some
degree, right?

00:04:02.210 --> 00:04:03.770
That provides some utility.

00:04:03.770 --> 00:04:05.020
It makes us happier.

00:04:07.140 --> 00:04:09.690
Or it could be things that help
produce these things like

00:04:09.690 --> 00:04:13.470
factories or-- who knows what
else-- computers, things that

00:04:13.470 --> 00:04:17.000
help us produce all of these
things more efficiently.

00:04:17.000 --> 00:04:20.630
And if you stay in that mindset,
money is not wealth.

00:04:20.630 --> 00:04:22.950
Money is just a medium of
exchange for wealth.

00:04:30.660 --> 00:04:31.720
You could say money has value.

00:04:31.720 --> 00:04:36.200
The value of money really is the
oil or the lubrication of

00:04:36.200 --> 00:04:36.730
the economy.

00:04:36.730 --> 00:04:42.060
Without any form of money, you
would have to be bartering.

00:04:42.060 --> 00:04:45.640
If I was a a home builder and I
wanted to get bread, I would

00:04:45.640 --> 00:04:48.560
have to exchange a house for, I
don't know, 10,000 loaves of

00:04:48.560 --> 00:04:50.210
bread, which just is
not practical.

00:04:50.210 --> 00:04:52.410
Instead, I can exchange it for
money and then later exchange

00:04:52.410 --> 00:04:54.330
that money for bread.

00:04:54.330 --> 00:04:57.790
And within this money medium of
exchange context, there's

00:04:57.790 --> 00:04:59.070
different types of
money, right?

00:04:59.070 --> 00:05:00.460
There's paper money.

00:05:05.040 --> 00:05:08.880
There's hard currency,
et cetera.

00:05:08.880 --> 00:05:13.790
And gold would fall under
this context.

00:05:13.790 --> 00:05:17.450
And then there's a fundamental
question: Is gold

00:05:17.450 --> 00:05:23.410
fundamentally, intrinsically,
worth more than paper money?

00:05:23.410 --> 00:05:25.860
And I guess it goes to
just, what do you

00:05:25.860 --> 00:05:27.230
think is going to happen?

00:05:27.230 --> 00:05:31.030
I mean, the benefit or the
negative of gold or a lot of

00:05:31.030 --> 00:05:32.770
hard currencies is
that it's scarce.

00:05:35.910 --> 00:05:40.460
And by it being scarce, it's
not like you can just print

00:05:40.460 --> 00:05:42.200
this stuff and there's
always going to be a

00:05:42.200 --> 00:05:43.440
set supply of it.

00:05:43.440 --> 00:05:48.380
And so if someone took a
negative spin on it, scarcity

00:05:48.380 --> 00:05:49.630
would mean less flexible.

00:05:54.240 --> 00:05:56.950
Or if you took a positive spin
on it, you could say it can't

00:05:56.950 --> 00:05:58.200
be manipulated.

00:06:02.420 --> 00:06:06.780
And both of these are probably
true to some degree, right?

00:06:06.780 --> 00:06:08.130
But that's what it is.

00:06:08.130 --> 00:06:10.740
It's just a scarce
unit of exchange.

00:06:10.740 --> 00:06:13.870
And so I'm not saying that gold
will one day lose its

00:06:13.870 --> 00:06:17.300
ability to control some of these
things or be exchanged

00:06:17.300 --> 00:06:18.580
for some of these things.

00:06:18.580 --> 00:06:21.420
I'm just saying that money or
the unit of exchange is

00:06:21.420 --> 00:06:23.420
fundamentally different than
these things that actually

00:06:23.420 --> 00:06:24.320
provide utility.

00:06:24.320 --> 00:06:26.670
I mean, gold does have
some applications.

00:06:26.670 --> 00:06:28.130
Maybe you say it provides
entertainment

00:06:28.130 --> 00:06:29.650
in the form of jewelry.

00:06:29.650 --> 00:06:31.590
Maybe does provide
some utility.

00:06:31.590 --> 00:06:34.010
It can be used as fillings
or in some industrial or

00:06:34.010 --> 00:06:35.680
electronic applications,
but primarily

00:06:35.680 --> 00:06:37.010
it's a unit of exchange.

00:06:37.010 --> 00:06:40.420
And as a unit of exchange, it
helps just facilitate the

00:06:40.420 --> 00:06:45.270
transactions amongst wealth and
helps the economy allocate

00:06:45.270 --> 00:06:46.780
resources efficiently.

00:06:46.780 --> 00:06:49.470
Now, paper money is
not scarce, right?

00:06:49.470 --> 00:06:50.720
It's controllable.

00:06:54.610 --> 00:06:56.590
And so you could take a negative
or positive spin on

00:06:56.590 --> 00:06:57.320
that, right?

00:06:57.320 --> 00:06:59.730
You could say it's flexible.

00:06:59.730 --> 00:07:02.300
That would be the positive
spin on it.

00:07:02.300 --> 00:07:05.420
Or you could say it could
be manipulated.

00:07:05.420 --> 00:07:10.540
And there are times in history
where many governments, just

00:07:10.540 --> 00:07:16.580
kind of push for populist
agendas or to somehow try to

00:07:16.580 --> 00:07:20.870
defer problems maybe until the
next guy shows up, they did

00:07:20.870 --> 00:07:21.660
manipulate currencies.

00:07:21.660 --> 00:07:23.910
And that did lead to
hyperinflation and eventually

00:07:23.910 --> 00:07:26.190
the paper money becomes
worthless, but then there's

00:07:26.190 --> 00:07:29.040
other examples where the paper
money, for the most part,

00:07:29.040 --> 00:07:31.650
really does retain its value.

00:07:31.650 --> 00:07:34.450
So that's another thing
to think about.

00:07:34.450 --> 00:07:38.200
And so just going back to the
question of whether gold will

00:07:38.200 --> 00:07:41.070
or will not lose its value,
that's just really-- I mean,

00:07:41.070 --> 00:07:44.340
what is it going to lose its
value relative to and what

00:07:44.340 --> 00:07:45.510
would be the catalyst for it?

00:07:45.510 --> 00:07:47.520
So a lot of people right now,
they'll say, the dollar--

00:07:47.520 --> 00:07:49.170
there's going to be
hyperinflation because the Fed

00:07:49.170 --> 00:07:53.440
is being so proactive, so
I'm going to buy gold.

00:07:53.440 --> 00:07:54.610
And they might be right.

00:07:54.610 --> 00:07:55.440
I'm not sure about that.

00:07:55.440 --> 00:07:56.730
If I knew about that
I would go and

00:07:56.730 --> 00:07:59.110
speculate on gold myself.

00:07:59.110 --> 00:08:05.170
But the main thing in any type
of investment is to realize

00:08:05.170 --> 00:08:08.120
that you're making a bet on
what human beings will do,

00:08:08.120 --> 00:08:09.530
what the Federal government
will do.

00:08:09.530 --> 00:08:13.640
And just as an example, in real
terms, if you adjust it

00:08:13.640 --> 00:08:23.130
for today's dollars, gold
in 1980 was-- this is in

00:08:23.130 --> 00:08:26.650
inflation adjusted dollars-- I
think it was on the order of

00:08:26.650 --> 00:08:28.450
$2,400, right?

00:08:28.450 --> 00:08:31.520
So even today, even with all
of this talk-- I guess it's

00:08:31.520 --> 00:08:35.210
2009 now-- even with all this
talk about hyperinflation, the

00:08:35.210 --> 00:08:39.620
Fed being overly aggressive and
us pumping up the money

00:08:39.620 --> 00:08:42.750
supply and the Fed's balance
sheet expanding-- even today,

00:08:42.750 --> 00:08:46.690
gold-- in today's terms-- this
is $2,400 in today's terms. I

00:08:46.690 --> 00:08:49.440
think it's at $840
or something.

00:08:49.440 --> 00:08:55.280
And back in 1980, a very
reasonable person could have

00:08:55.280 --> 00:08:57.530
said, my God, we have
hyperinflation, we just got

00:08:57.530 --> 00:09:00.330
through these oil shocks,
inflation is going through the

00:09:00.330 --> 00:09:02.980
roof-- inflation was already
much more rampant than it has

00:09:02.980 --> 00:09:06.320
been in recent memory-- and
you're like, boy, the dollar's

00:09:06.320 --> 00:09:09.370
going to be worthless
before we know it.

00:09:09.370 --> 00:09:12.070
I'm just going to buy gold,
although-- what

00:09:12.070 --> 00:09:12.980
should gold be worth?

00:09:12.980 --> 00:09:14.860
Should it be worth $24,000?

00:09:14.860 --> 00:09:17.130
Should it be worth $2 million?

00:09:17.130 --> 00:09:18.290
Who knows?

00:09:18.290 --> 00:09:20.850
Does gold just go to infinity
in terms of dollars or does

00:09:20.850 --> 00:09:22.370
the dollar just go to zero?

00:09:22.370 --> 00:09:24.790
So someone could've made a very
reasonable argument back

00:09:24.790 --> 00:09:28.910
then that they should get out
of their bank accounts and

00:09:28.910 --> 00:09:31.240
convert it all to gold and they
wouldn't have been able

00:09:31.240 --> 00:09:34.110
to predict that Paul Volcker
would show up in the early

00:09:34.110 --> 00:09:37.510
'80s and increase interest
rates despite all of the

00:09:37.510 --> 00:09:41.380
political pressure to maybe
do the opposite and he got

00:09:41.380 --> 00:09:42.410
inflation under control.

00:09:42.410 --> 00:09:43.870
And then by-- what was it?

00:09:48.460 --> 00:09:52.190
By the late '90s, in today's
terms, gold was-- I think in

00:09:52.190 --> 00:09:54.420
the $200 range or it might've
been even lower.

00:09:54.420 --> 00:09:57.280
We'll say approximately $200.

00:09:57.280 --> 00:09:59.850
And even if that person said,
even in the late '90s, I'm

00:09:59.850 --> 00:10:04.480
going to hold to my bet and keep
holding gold, it would've

00:10:04.480 --> 00:10:07.270
taken-- if they'd had it all the
way to today, they still

00:10:07.270 --> 00:10:10.110
would have been better off in
1980 keeping their money in a

00:10:10.110 --> 00:10:12.350
bank account that gets more
interest than inflation.

00:10:12.350 --> 00:10:15.530
So if you just got 3% or 4% on
your bank account starting in

00:10:15.530 --> 00:10:18.250
1980, you still would have been
better off than holding

00:10:18.250 --> 00:10:21.420
gold-- even though there was a
very credible argument for

00:10:21.420 --> 00:10:22.270
inflation at that time.

00:10:22.270 --> 00:10:24.360
So it's just a very, very
hard thing to predict.

00:10:24.360 --> 00:10:26.470
And with any of these things
where you're really just

00:10:26.470 --> 00:10:28.460
betting on the medium of
exchange, it's very

00:10:28.460 --> 00:10:29.200
hard to price it.

00:10:29.200 --> 00:10:31.650
It's very hard to say
what that unit of

00:10:31.650 --> 00:10:33.580
exchange really is.

00:10:33.580 --> 00:10:37.920
So now gold has ran up from
the late '90s to $840.

00:10:37.920 --> 00:10:41.420
It's based on a lot of people
thinking that the dollar's

00:10:41.420 --> 00:10:43.110
going to lose its value,
et cetera.

00:10:43.110 --> 00:10:44.260
It's completely possible.

00:10:44.260 --> 00:10:47.000
We might have an irresponsible
government.

00:10:47.000 --> 00:10:48.920
Inflation looks like it's under
control right now, but

00:10:48.920 --> 00:10:50.670
who knows what happens
in a year or two?

00:10:50.670 --> 00:10:52.670
Maybe all this liquidity starts
to get a multiplier

00:10:52.670 --> 00:10:55.460
effect and the dollar
gets debased and

00:10:55.460 --> 00:10:58.850
gold jumps up to $8,000.

00:10:58.850 --> 00:11:02.140
Then those people just have made
a great bet, but maybe

00:11:02.140 --> 00:11:03.170
inflation isn't the problem.

00:11:03.170 --> 00:11:05.210
Maybe it's deflation and
maybe people don't lose

00:11:05.210 --> 00:11:06.030
faith in the currency.

00:11:06.030 --> 00:11:11.250
Maybe a Paul Volcker-like
character shows up and tackles

00:11:11.250 --> 00:11:14.610
inflation and then gold
goes back to $200.

00:11:14.610 --> 00:11:17.340
But the bottom line here is
that it's really hard to

00:11:17.340 --> 00:11:21.080
predict because it's actually
not generating utility itself,

00:11:21.080 --> 00:11:22.730
it's a measure of utility.

00:11:22.730 --> 00:11:24.430
But anyway, I don't want to
fixate on that too much, but

00:11:24.430 --> 00:11:26.370
it's an interesting
discussion.

00:11:26.370 --> 00:11:28.080
But anyway, just going back to
the original thing, there's

00:11:28.080 --> 00:11:30.980
other ways that we could have
gone down the evolutionary

00:11:30.980 --> 00:11:33.080
path of banking systems. I just
wanted to go over the

00:11:33.080 --> 00:11:35.550
path that we've gone.

00:11:35.550 --> 00:11:37.170
Instead of a fractional reserve,
you could have a full

00:11:37.170 --> 00:11:38.420
reserve system.

00:11:42.340 --> 00:11:45.230
That's-- instead of me taking a
deposit and saying that you

00:11:45.230 --> 00:11:47.840
can demand the deposit at any
given point, but still lending

00:11:47.840 --> 00:11:50.170
90% of it out-- that's what I do
in the fractional reserve--

00:11:50.170 --> 00:11:52.600
I would tell you, 90%
of your money, you

00:11:52.600 --> 00:11:53.620
can't take it out tomorrow.

00:11:53.620 --> 00:11:55.770
It's going to be locked
in for six months.

00:11:55.770 --> 00:11:58.610
It'd be like a CD and the money
that you want to have

00:11:58.610 --> 00:12:01.100
tomorrow, I'm just going to put
it aside, you're not going

00:12:01.100 --> 00:12:03.040
to get any interest on it, and
you might have to even pay me

00:12:03.040 --> 00:12:05.660
a little bit of money to keep
it safe and to be able to

00:12:05.660 --> 00:12:06.780
access it from an ATM.

00:12:06.780 --> 00:12:08.600
That's completely fine.

00:12:08.600 --> 00:12:11.870
Someone could argue that
this is less flexible.

00:12:11.870 --> 00:12:15.720
That would be the negative
spin on it.

00:12:15.720 --> 00:12:20.580
But then the positive spin on it
is maybe it's less subject

00:12:20.580 --> 00:12:21.830
to manipulation.

00:12:23.960 --> 00:12:27.910
Or maybe there will be a few
less booms and busts, right?

00:12:27.910 --> 00:12:30.020
Maybe these bubbles
won't form.

00:12:30.020 --> 00:12:32.210
I'm not so sure about that,
but I think it's

00:12:32.210 --> 00:12:33.190
a reasonable argument.

00:12:33.190 --> 00:12:35.230
And then even within full
reserve, we could have gone

00:12:35.230 --> 00:12:38.890
gold or not gold.

00:12:38.890 --> 00:12:40.520
And once again, that would
have just been a question

00:12:40.520 --> 00:12:44.320
about how much flexibility you
have. The most restrictive

00:12:44.320 --> 00:12:47.310
system would be a full reserve
system based on some type of

00:12:47.310 --> 00:12:51.170
hard currency that can't be
printed or can't be controlled

00:12:51.170 --> 00:12:52.860
by humans to some degree.

00:12:52.860 --> 00:12:54.940
Or you could have something that
is printed and then to

00:12:54.940 --> 00:12:58.510
some degree this could be
manipulated a little bit.

00:12:58.510 --> 00:13:00.240
And just like on the fractional
reserve--

00:13:00.240 --> 00:13:02.520
fractional reserve backed by
gold-- little less flexible,

00:13:02.520 --> 00:13:03.270
but maybe that's good.

00:13:03.270 --> 00:13:05.240
Little less game-able as well.

00:13:05.240 --> 00:13:08.160
Anyway, I just wanted to give
you this kind of overview of

00:13:08.160 --> 00:13:08.560
everything.

00:13:08.560 --> 00:13:11.700
Because I think I maybe struck
some nerves in the last time.

00:13:11.700 --> 00:13:15.030
I didn't want to in any way
insult gold and I don't think

00:13:15.030 --> 00:13:17.830
that tomorrow gold will
be worthless.

00:13:17.830 --> 00:13:20.190
I just wanted to make it
very clear that from a

00:13:20.190 --> 00:13:23.840
philosophical point of view,
there's a difference between

00:13:23.840 --> 00:13:27.000
gold and the things that it can
be exchanged for, just as

00:13:27.000 --> 00:13:29.240
there is a difference between
money or-- any kind of money--

00:13:29.240 --> 00:13:31.680
paper money and the things that
it can be exchanged for.

00:13:31.680 --> 00:13:35.410
And in general, I think gold
does have a special place

00:13:35.410 --> 00:13:37.160
relative to any other
hard currencies.

00:13:37.160 --> 00:13:42.400
Relative to diamonds, or
nanotubes, or anything else--

00:13:42.400 --> 00:13:45.340
and that's just because it
has a history, right?

00:13:45.340 --> 00:13:50.020
From the beginning of recorded
history, people used gold as a

00:13:50.020 --> 00:13:52.040
medium of exchange, so it's
almost ingrained in our

00:13:52.040 --> 00:13:55.690
cultural DNA and that's what
gives it its robustness as a

00:13:55.690 --> 00:13:58.490
unit of exchange, but other
than that, it's really no

00:13:58.490 --> 00:14:00.110
different than diamonds,
or carbon

00:14:00.110 --> 00:14:02.170
nanotubes, or moon rocks.

00:14:02.170 --> 00:14:04.480
Anyway, I've used
up all my time.

00:14:04.480 --> 00:14:06.520
I'll see you in the
next video.


CA Geometry: Secants and Translations

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.190
We're on 76.

00:00:02.190 --> 00:00:07.140
In the figure below, line
AB is tangent to

00:00:07.140 --> 00:00:09.000
circle O at point A.

00:00:09.000 --> 00:00:11.240
Fair enough.

00:00:11.240 --> 00:00:16.040
Secant BD intersects circle
O at points C and D.

00:00:19.370 --> 00:00:20.700
Secant just means that
it intersects the

00:00:20.700 --> 00:00:21.800
point at two points.

00:00:21.800 --> 00:00:22.590
It's not tangent.

00:00:22.590 --> 00:00:25.630
A tangent intersects the circle
at exactly one point.

00:00:25.630 --> 00:00:30.020
Secant BD intersects the circle
at points C and D.

00:00:30.020 --> 00:00:38.850
Measure of arc AC is equal to 70
degrees, and the measure of

00:00:38.850 --> 00:00:43.800
arc CD is equal to
110 degrees.

00:00:43.800 --> 00:00:45.690
So what do they want
us to figure out?

00:00:45.690 --> 00:00:46.940
What is the measure of ABC?

00:00:50.600 --> 00:00:51.650
So what is this angle?

00:00:51.650 --> 00:00:53.930
That's what they want
us to figure out.

00:00:53.930 --> 00:00:58.080
They want us to figure
out this right here.

00:00:58.080 --> 00:00:59.490
That angle.

00:00:59.490 --> 00:01:01.690
Let's see what we can do here.

00:01:01.690 --> 00:01:04.920
So there's a couple of ways
to think about it.

00:01:04.920 --> 00:01:11.940
Well, if you look at it, this
is 70, this is 110, right?

00:01:11.940 --> 00:01:15.210
So if you add them, this whole
arc, going for all the way

00:01:15.210 --> 00:01:20.320
around, that's 180 degrees.

00:01:20.320 --> 00:01:23.270
So it's actually perfectly
halfway around the circle.

00:01:23.270 --> 00:01:25.380
So actually from A to
D is actually a

00:01:25.380 --> 00:01:27.010
diameter of the circle.

00:01:27.010 --> 00:01:27.760
How do I know that?

00:01:27.760 --> 00:01:31.170
Because it goes all the
way around the circle.

00:01:31.170 --> 00:01:33.925
The arc length is 180 degrees.

00:01:33.925 --> 00:01:37.180
So we can draw-- let me
draw a diameter line.

00:01:37.180 --> 00:01:38.710
I don't know if this thing
is drawn to scale.

00:01:41.920 --> 00:01:44.230
A to D is actually a diameter
of the circle.

00:01:44.230 --> 00:01:46.510
We know that, because that's 180
degrees, the combined arc

00:01:46.510 --> 00:01:48.090
length right there.

00:01:48.090 --> 00:01:51.535
And we know, that this line
is tangent at point A.

00:01:56.020 --> 00:01:59.180
So that has to be
a right angle.

00:01:59.180 --> 00:02:02.050
When you're tangent at point A,
that means your tangent to

00:02:02.050 --> 00:02:03.580
the radius at that point.

00:02:03.580 --> 00:02:04.750
Fair enough.

00:02:04.750 --> 00:02:06.030
So now let's see what
we can figure out.

00:02:06.030 --> 00:02:07.880
We know that this arc
length is 70.

00:02:07.880 --> 00:02:12.640
Is there any way to figure out
what this angle right here is?

00:02:12.640 --> 00:02:14.710
Because if we know this angle,
we know this is 90, then we

00:02:14.710 --> 00:02:16.240
could figure out that angle.

00:02:16.240 --> 00:02:20.790
And this is something that you
should just learn about

00:02:20.790 --> 00:02:22.730
inscribed angles in a circle.

00:02:22.730 --> 00:02:26.670
This is an inscribed angle,
because the vertex touches on

00:02:26.670 --> 00:02:28.460
one of the sides
of the circle.

00:02:28.460 --> 00:02:31.700
So an inscribed angle-- and this
is just something good to

00:02:31.700 --> 00:02:33.640
memorize, and you could play
around with it to get a little

00:02:33.640 --> 00:02:37.630
bit more intuition about it--
is equal to half of the arc

00:02:37.630 --> 00:02:39.320
length that it intersects.

00:02:39.320 --> 00:02:42.790
So this inscribed angle
intersects an arc

00:02:42.790 --> 00:02:45.350
length of 70 degrees.

00:02:45.350 --> 00:02:50.410
So this is 35 degrees.

00:02:50.410 --> 00:02:51.830
And that's just something
good to know.

00:02:51.830 --> 00:02:53.190
That this is going to
be half of whatever

00:02:53.190 --> 00:02:55.510
this arc length is.

00:02:55.510 --> 00:03:03.370
Now we can figure out x, because
x plus 35 plus 90 is

00:03:03.370 --> 00:03:06.030
equal to 180.

00:03:06.030 --> 00:03:14.970
You get x plus 35 is equal to
90. x is equal to 90 minus 35

00:03:14.970 --> 00:03:20.660
is 55, which is choice C.

00:03:20.660 --> 00:03:22.470
Now there's another way you
can do this, and this is

00:03:22.470 --> 00:03:26.800
another interesting thing
about lines that

00:03:26.800 --> 00:03:28.710
intersect in circles.

00:03:28.710 --> 00:03:32.800
It's that this angle measure,
right here, if it's sitting

00:03:32.800 --> 00:03:36.470
outside of the circle-- I think
there might be ones

00:03:36.470 --> 00:03:38.440
sitting inside the circle later
on-- but if it's sitting

00:03:38.440 --> 00:03:44.120
outside of the circle, it's
equal to 1/2 the difference of

00:03:44.120 --> 00:03:46.970
the arc measures that
it intersects.

00:03:46.970 --> 00:03:50.980
So for example, it intersects
this arc measure of 70, and it

00:03:50.980 --> 00:03:53.940
also has this arc measure
right here.

00:03:57.250 --> 00:04:00.485
When it hits the circle and
intersects at 70 degrees and

00:04:00.485 --> 00:04:02.430
when it exits the circle,
it has this arc measure.

00:04:02.430 --> 00:04:04.150
This arc measure we already
figured out.

00:04:04.150 --> 00:04:07.290
It's 180 degrees, because
that's 180 degrees.

00:04:07.290 --> 00:04:09.850
So you know that this angle
right here, you could say that

00:04:09.850 --> 00:04:14.100
x is equal to 1/2 the difference
of these two arc

00:04:14.100 --> 00:04:16.149
measures, and we can do
that because x is

00:04:16.149 --> 00:04:17.160
outside of the circle.

00:04:17.160 --> 00:04:21.100
If x was inside the circle, then
x would be 1/2 the sum of

00:04:21.100 --> 00:04:21.720
the two measures.

00:04:21.720 --> 00:04:22.640
So what's the difference?

00:04:22.640 --> 00:04:26.670
It's 180 minus 70.

00:04:26.670 --> 00:04:32.910
So that's equal to 1/2 times
110, so that is equal to 55.

00:04:32.910 --> 00:04:37.210
So it's nice to see that math
works correctly no matter how

00:04:37.210 --> 00:04:39.190
you do the problem.

00:04:39.190 --> 00:04:44.830
Problem 77: In the circle shown
below, the measure of

00:04:44.830 --> 00:04:48.730
arc PR is 140 degrees.

00:04:48.730 --> 00:04:49.710
Sure enough.

00:04:49.710 --> 00:04:53.530
The measure of angle
RPQ is 50 degrees.

00:04:53.530 --> 00:04:55.410
OK, that's 50 degrees.

00:04:55.410 --> 00:04:59.360
What is the measure of arc PQ?

00:04:59.360 --> 00:05:02.700
What is this arc right there?

00:05:02.700 --> 00:05:05.950
So we could use what we learned
in the last problem.

00:05:05.950 --> 00:05:09.760
50 degrees is an inscribed
angle.

00:05:09.760 --> 00:05:13.860
So it's going to be half of
the arc length measured in

00:05:13.860 --> 00:05:17.810
degrees, or arc angle, I guess
you could say, of the arc that

00:05:17.810 --> 00:05:19.000
it intercepts.

00:05:19.000 --> 00:05:23.880
So this is 50, then this
arc measure is

00:05:23.880 --> 00:05:27.820
going to be 100 degrees.

00:05:27.820 --> 00:05:30.630
So what we're trying to figure
out, this arc measure right

00:05:30.630 --> 00:05:35.960
here, it's whatever you
really have left over.

00:05:35.960 --> 00:05:36.880
Let's call this x.

00:05:36.880 --> 00:05:45.570
So we know that x plus 100 plus
140, well, that goes all

00:05:45.570 --> 00:05:47.590
the way around the
circle, right?

00:05:47.590 --> 00:05:50.310
That's going to be equal
to 360 degrees.

00:05:50.310 --> 00:05:55.120
So you have x plus 240 is
equal to 360 degrees.

00:05:55.120 --> 00:05:58.130
Subtract 240 from both sides,
and you get x is

00:05:58.130 --> 00:06:02.080
equal to 120 degrees.

00:06:02.080 --> 00:06:03.330
Choice D.

00:06:05.600 --> 00:06:12.430
Problem 78: The vertices of
triangle ABC are A is 2 comma

00:06:12.430 --> 00:06:16.370
1, B is 3 comma 4, and
C is 1 comma 3.

00:06:16.370 --> 00:06:20.310
If triangle ABC is translated
one unit down-- so one unit

00:06:20.310 --> 00:06:23.890
down means that we're
subtracting 1 from the y's--

00:06:23.890 --> 00:06:26.340
and three units to the left--
so that means you're

00:06:26.340 --> 00:06:31.250
subtracting 3 from the x's-- to
create triangle DEF, what

00:06:31.250 --> 00:06:33.270
are the coordinates of
the vertices of DEF?

00:06:33.270 --> 00:06:37.290
So let's see, we have
A is 2 comma 1.

00:06:48.020 --> 00:06:51.830
And that's going to be
translated to point D.

00:06:51.830 --> 00:06:52.330
What do they say?

00:06:52.330 --> 00:06:53.570
One unit down.

00:06:53.570 --> 00:06:56.270
One unit down means the
y decreases by 1.

00:06:56.270 --> 00:06:58.500
So the y-coordinate's going
to be-- decrease that

00:06:58.500 --> 00:07:01.180
by 1, you get 0.

00:07:01.180 --> 00:07:03.300
And three units to the left,
that means you decrease the

00:07:03.300 --> 00:07:05.190
x-coordinate by 3.

00:07:05.190 --> 00:07:10.260
So 2 minus 3 is negative one.

00:07:10.260 --> 00:07:13.170
Actually, just looking at the
choices, we're already done.

00:07:13.170 --> 00:07:16.150
D is the only choice that had
a negative 1 comma 0.

00:07:16.150 --> 00:07:17.530
But let's look at
the other ones.

00:07:17.530 --> 00:07:20.120
Maybe somehow they changed the
lettering or something.

00:07:20.120 --> 00:07:24.916
Let's see point B
was 3 comma 4.

00:07:28.230 --> 00:07:29.290
So what do we do?

00:07:29.290 --> 00:07:31.850
With x, we said three units
to the left, so that

00:07:31.850 --> 00:07:35.580
becomes point E.

00:07:35.580 --> 00:07:38.020
So you take this three
units to left.

00:07:38.020 --> 00:07:40.630
You're subtracting 3 from
there, that becomes 0.

00:07:40.630 --> 00:07:44.170
If you take the y-coordinate
one unit down,

00:07:44.170 --> 00:07:45.260
that becomes a 3.

00:07:45.260 --> 00:07:47.000
So E is 0 comma 3.

00:07:47.000 --> 00:07:48.270
So that's still consistent
with this.

00:07:48.270 --> 00:07:50.300
And let's just make sure
that the F works.

00:07:50.300 --> 00:07:52.625
So C was 1 comma 3.

00:07:55.690 --> 00:08:00.530
If we take 3 away, we shift it
to the left by 3, so that's

00:08:00.530 --> 00:08:03.110
taking 3 away from the
x-coordinate, so

00:08:03.110 --> 00:08:04.590
that's minus 2.

00:08:04.590 --> 00:08:07.320
And if you're shifting the
y-coordinate down one, that's

00:08:07.320 --> 00:08:10.220
taking 1 from it, so that's 3,
so F is minus 2 comma 2.

00:08:10.220 --> 00:08:14.090
And so D was definitely
the answer.

00:08:14.090 --> 00:08:20.460
Problem 79: If triangle ABC is
rotated 180 degrees about the

00:08:20.460 --> 00:08:24.640
origin, what are the coordinates
of A prime?

00:08:24.640 --> 00:08:30.390
OK, so we're going to rotate
this thing 180 degrees.

00:08:30.390 --> 00:08:32.390
And essentially, we can worry
about all the coordinates, but

00:08:32.390 --> 00:08:36.270
they just want to know where A
prime is, so wherever A sits

00:08:36.270 --> 00:08:38.220
after we've rotated it.

00:08:38.220 --> 00:08:41.650
So let's think about
it a little bit.

00:08:41.650 --> 00:08:45.370
When we've rotated the C, this
is going to be the new C.

00:08:45.370 --> 00:08:46.620
It's going to be right there.

00:08:52.020 --> 00:08:56.130
B is going to be, when you
rotate it-- I just want to

00:08:56.130 --> 00:08:57.330
make sure I'm doing
this right.

00:08:57.330 --> 00:09:00.870
This is almost like a
visualization problem.

00:09:00.870 --> 00:09:12.310
If you go four to the
right and up three.

00:09:12.310 --> 00:09:16.190
So if you were to rotate
it around.

00:09:16.190 --> 00:09:18.710
Let's say this little arrow I
drew with the four, if I were

00:09:18.710 --> 00:09:23.450
to rotate that 180 degrees-- I
just want to make sure I do it

00:09:23.450 --> 00:09:26.850
right-- then that's going
to be four this way.

00:09:31.200 --> 00:09:34.740
So you're going to
be at that point.

00:09:34.740 --> 00:09:37.330
And then instead of three up,
you're going to go three down.

00:09:40.760 --> 00:09:42.950
So that's where A prime
is going to be.

00:09:42.950 --> 00:09:45.790
Right at that point.

00:09:45.790 --> 00:09:49.010
And that's the point minus
five comma minus four.

00:09:52.720 --> 00:09:53.440
And that's our answer.

00:09:53.440 --> 00:09:55.020
But let's figure out all
of the other points.

00:09:55.020 --> 00:10:01.360
So B, you go up three and
to the right two.

00:10:18.570 --> 00:10:23.870
The new B, you'd go three
down to the left.

00:10:23.870 --> 00:10:24.560
OK, it would be there.

00:10:24.560 --> 00:10:25.530
So that's the new B.

00:10:25.530 --> 00:10:25.870
Right.

00:10:25.870 --> 00:10:27.370
And so that looks right.

00:10:27.370 --> 00:10:29.310
That's what the triangle
is going to look like.

00:10:29.310 --> 00:10:34.170
This is B prime, this is A
prime, this is C prime.

00:10:34.170 --> 00:10:37.010
This is the new triangle when
you rotate it 180 degrees.

00:10:37.010 --> 00:10:40.040
That was a challenging
visualization problem, but the

00:10:40.040 --> 00:10:42.620
key thing is just to
keep visualizing

00:10:42.620 --> 00:10:43.900
going around 180 degrees.

00:10:43.900 --> 00:10:46.660
So minus five comma
minus four, that's

00:10:46.660 --> 00:10:50.160
choice A for A prime.

00:10:50.160 --> 00:10:58.930
Problem 80: Trapezoid ABCD below
is to be translated to

00:10:58.930 --> 00:11:01.750
trapezoid A prime, B prime,
C prime, D prime by the

00:11:01.750 --> 00:11:05.610
following motion rule.

00:11:05.610 --> 00:11:08.000
What will be the coordinates
of vertex C prime?

00:11:08.000 --> 00:11:10.780
OK, so that's all we have
to worry about.

00:11:10.780 --> 00:11:13.100
We're at C and we're going to
translate it to some C prime.

00:11:13.100 --> 00:11:14.350
What of the coordinates of C?

00:11:17.010 --> 00:11:22.220
x is equal to five, and
y is equal to 1.

00:11:22.220 --> 00:11:24.210
So what's C prime going
to be equal to?

00:11:24.210 --> 00:11:27.280
C prime, we'll just use
this mapping rule.

00:11:27.280 --> 00:11:31.290
You add 3 to the x, so it's
going to be 8, and you

00:11:31.290 --> 00:11:34.000
subtract 4 from the y.

00:11:34.000 --> 00:11:36.600
1 minus 4 is minus 3.

00:11:36.600 --> 00:11:38.980
8 minus 3.

00:11:38.980 --> 00:11:41.880
That's choice D.

00:11:41.880 --> 00:11:42.610
And we're all done.

00:11:42.610 --> 00:11:45.195
We're all done with the entire
California Standards geometry

00:11:45.195 --> 00:11:48.870
test. And frankly, I started
off grumbling a little bit,

00:11:48.870 --> 00:11:50.300
because I thought they were
getting a little bit too

00:11:50.300 --> 00:11:51.740
obsessed with the terminology.

00:11:51.740 --> 00:11:54.730
But overall, I think this was a
pretty good test. And if you

00:11:54.730 --> 00:11:58.790
understood every problem on this
exam, I think you know

00:11:58.790 --> 00:12:00.670
your geometry pretty well.

00:12:00.670 --> 00:12:03.930
Anyway, see you in the next
video, although this is the

00:12:03.930 --> 00:12:07.380
last in this geometry series.


CA Geometry: Circle area chords tangent

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.790
We're on problem 71.

00:00:02.790 --> 00:00:06.830
It says, what is the value of
x in the triangle below?

00:00:06.830 --> 00:00:10.360
OK, so we can pull out the
Pythagorean theorem here.

00:00:10.360 --> 00:00:13.390
Although, you might recognize
that if these two sides are

00:00:13.390 --> 00:00:15.480
the same, then these
two base angles are

00:00:15.480 --> 00:00:16.630
going to be the same.

00:00:16.630 --> 00:00:21.080
And if those base angles are
the same, then you have 90

00:00:21.080 --> 00:00:23.045
degrees to go between those two
angles, so they're going

00:00:23.045 --> 00:00:24.110
to have to be 45.

00:00:24.110 --> 00:00:25.360
Because they're the same.

00:00:25.360 --> 00:00:29.080
So this is a 45, 45,
90 triangle.

00:00:29.080 --> 00:00:32.430
And if you haven't already,
you'll eventually memorize how

00:00:32.430 --> 00:00:35.735
the sides of a 45, 45, 90 relate
to its hypotenuse.

00:00:35.735 --> 00:00:36.980
But you don't have
to memorize.

00:00:36.980 --> 00:00:37.710
You can prove it here.

00:00:37.710 --> 00:00:39.480
Sometimes it's just faster
on standardized tests

00:00:39.480 --> 00:00:40.530
and things like that.

00:00:40.530 --> 00:00:42.370
So what does the Pythagorean
theorem tell us?

00:00:42.370 --> 00:00:45.600
It tells us that this side
squared, so let's say x

00:00:45.600 --> 00:00:49.660
squared, plus this side squared,
plus x squared is

00:00:49.660 --> 00:00:51.370
equal to the hypotenuse
squared.

00:00:51.370 --> 00:00:54.890
Is equal to 10 squared,
which is 100.

00:00:54.890 --> 00:00:58.980
So we get 2x squared
is equal to 100.

00:00:58.980 --> 00:01:03.380
x squared is equal to 50.

00:01:03.380 --> 00:01:05.480
Dividing both sides by 2.

00:01:05.480 --> 00:01:06.830
And then what does
this turn into?

00:01:06.830 --> 00:01:13.270
So we can say x is equal to
the square root of 50.

00:01:13.270 --> 00:01:18.140
Is there anything that we can do
here to simplify it at all.

00:01:18.140 --> 00:01:21.060
Let me think.

00:01:21.060 --> 00:01:28.090
Oh sure, 50 is 25 times 2.

00:01:28.090 --> 00:01:31.150
So that's equal to the square
root of 25 times the

00:01:31.150 --> 00:01:32.490
square root of 2.

00:01:32.490 --> 00:01:36.760
Which is equal to 5 times
the square root of 2.

00:01:36.760 --> 00:01:40.210
Choice B.

00:01:40.210 --> 00:01:41.910
Problem 72.

00:01:41.910 --> 00:01:45.670
What is the value
of x in inches?

00:01:45.670 --> 00:01:49.690
OK, a couple of problems ago, we
saw a 30, 60, 90 triangle,

00:01:49.690 --> 00:01:50.770
this is another one.

00:01:50.770 --> 00:01:54.060
30 degrees, 90 degrees, they
have to add up to 180, this

00:01:54.060 --> 00:01:56.310
one is equal to 60 degrees.

00:01:56.310 --> 00:01:59.340
And I did that big convoluted
drawing where I flipped it and

00:01:59.340 --> 00:01:59.790
all of that.

00:01:59.790 --> 00:02:02.950
I think this is a good time to
just to memorize the sides of

00:02:02.950 --> 00:02:04.000
a 30, 60, 90 triangle.

00:02:04.000 --> 00:02:07.360
Because that's something that
one needs to know in life.

00:02:07.360 --> 00:02:08.500
It's surprisingly useful.

00:02:08.500 --> 00:02:10.830
Especially once you start taking
standardized tests or

00:02:10.830 --> 00:02:11.650
do trigonometry.

00:02:11.650 --> 00:02:13.440
So I'll just give you
the general rule.

00:02:13.440 --> 00:02:16.350
So let me just draw another
one right here.

00:02:16.350 --> 00:02:22.280
Let's say this is my other
30, 60, 90 triangle.

00:02:22.280 --> 00:02:25.800
This is clearly the hypotenuse
up here.

00:02:25.800 --> 00:02:29.930
This is, I call it the 30 degree
side, it's opposite the

00:02:29.930 --> 00:02:32.420
30 degree angle, or it's
the shortest side.

00:02:32.420 --> 00:02:37.290
So the general rule is, if this
side right here is x.

00:02:37.290 --> 00:02:39.670
Then the hypotenuse
is going to be 2x.

00:02:39.670 --> 00:02:41.820
And we saw that in that
previous video.

00:02:41.820 --> 00:02:44.700
And then you can actually use
the Pythagorean theorem here

00:02:44.700 --> 00:02:46.400
to solve for this last side.

00:02:46.400 --> 00:02:48.450
You really just need to memorize
that the hypotenuse

00:02:48.450 --> 00:02:50.760
is twice the shortest side.

00:02:50.760 --> 00:02:52.570
So this case, what's
the shortest side?

00:02:52.570 --> 00:02:54.160
It's opposite the
30 degree side.

00:02:54.160 --> 00:02:55.010
So it's 7.

00:02:55.010 --> 00:02:59.300
So the hypotenuse would be
twice that, which is 14.

00:02:59.300 --> 00:03:00.990
And you could use the
Pythagorean theorem to

00:03:00.990 --> 00:03:02.270
figure out x now.

00:03:02.270 --> 00:03:06.140
Or you could just memorize that
the middle side, I guess

00:03:06.140 --> 00:03:09.940
you could say, or the long
non-hypotenuse side, or the 60

00:03:09.940 --> 00:03:13.660
degree side, the side opposite
the 60 degree angle, that's

00:03:13.660 --> 00:03:19.120
equal to the square root of
3 times the short side.

00:03:19.120 --> 00:03:22.900
So in this case, x is the square
root of 3 times 7.

00:03:22.900 --> 00:03:28.910
So x is equal to 7 times
the square root of 3.

00:03:28.910 --> 00:03:30.190
And don't take my word for it.

00:03:30.190 --> 00:03:32.220
You could take my word for
that this is double that.

00:03:32.220 --> 00:03:34.280
And we proved that in a
couple of videos ago.

00:03:34.280 --> 00:03:35.690
But you could do the Pythagorean
theorem here.

00:03:35.690 --> 00:03:41.200
You could say that 7 squared,
which is 49 , plus x squared

00:03:41.200 --> 00:03:43.490
is going to be equal to the
hypotenuse squared.

00:03:43.490 --> 00:03:52.490
14 squared is 196.

00:03:52.490 --> 00:03:56.090
Subtract 49 from both sides.

00:03:56.090 --> 00:04:08.990
You get x squared is equal to
196 minus 50 would be 157, is

00:04:08.990 --> 00:04:11.390
that right?

00:04:11.390 --> 00:04:12.330
Let me make sure I got it.

00:04:12.330 --> 00:04:15.200
14 times 14.

00:04:15.200 --> 00:04:17.370
4 times 4 is 16.

00:04:17.370 --> 00:04:18.730
56.

00:04:18.730 --> 00:04:19.670
140, right.

00:04:19.670 --> 00:04:21.500
196.

00:04:21.500 --> 00:04:27.270
And if you were to subtract 49
from that, this is an 8, this

00:04:27.270 --> 00:04:31.060
is 16, we have a 7.

00:04:31.060 --> 00:04:33.280
Sorry, 147.

00:04:33.280 --> 00:04:34.530
It's a good thing
I checked that.

00:04:37.030 --> 00:04:38.050
All right.

00:04:38.050 --> 00:04:43.520
So x is equal to the
square root of 147.

00:04:43.520 --> 00:04:46.490
147 is 49 times 3.

00:04:46.490 --> 00:04:50.300
It's equal to the square
root of 49 times 3.

00:04:50.300 --> 00:04:54.270
Well that's just equal to the
square root of 49 times the

00:04:54.270 --> 00:04:55.720
square root of 3.

00:04:55.720 --> 00:04:58.270
Which is equal to 7 root 3.

00:04:58.270 --> 00:04:59.510
Which is what we got.

00:04:59.510 --> 00:05:02.650
But it might be easier to just
memorize that the side

00:05:02.650 --> 00:05:05.660
opposite the 60 degree side, is
going to be the square root

00:05:05.660 --> 00:05:07.380
3 times the short side.

00:05:07.380 --> 00:05:10.630
And the short side is going
to be half the hypotenuse.

00:05:10.630 --> 00:05:12.190
Anyway, the more practice
you do, the more

00:05:12.190 --> 00:05:13.840
it will make sense.

00:05:13.840 --> 00:05:17.860
OK, a square is circumscribed
about a circle.

00:05:17.860 --> 00:05:20.320
What is the ratio
of the circle to

00:05:20.320 --> 00:05:23.480
the area of the square?

00:05:23.480 --> 00:05:24.850
Let me draw the circle
and the square.

00:05:49.490 --> 00:05:51.220
Well, I think that's
close enough.

00:05:51.220 --> 00:05:52.910
We know the square is on
the outside because

00:05:52.910 --> 00:05:56.190
it's about the circle.

00:05:56.190 --> 00:05:58.510
What is the ratio of the area
of the circle to the area of

00:05:58.510 --> 00:06:00.120
the square?

00:06:00.120 --> 00:06:02.270
So let's say this is
the center of the

00:06:02.270 --> 00:06:03.940
circle right there.

00:06:03.940 --> 00:06:05.112
This is its radius.

00:06:05.112 --> 00:06:07.250
Let's call that r.

00:06:07.250 --> 00:06:09.910
Well what's the area of the
square going to be?

00:06:09.910 --> 00:06:13.540
If that's the radius, this
is also the radius.

00:06:13.540 --> 00:06:17.550
So one side of this square up
here, is going to be 2r.

00:06:17.550 --> 00:06:19.040
So this side is also
going to be 2r.

00:06:19.040 --> 00:06:21.960
It's a square, all the
sides are the same.

00:06:21.960 --> 00:06:25.600
So they want to know the ratio
of the area of the circle to

00:06:25.600 --> 00:06:26.850
the area of the square.

00:06:33.660 --> 00:06:36.390
The area of the square
is just 2r times 2r.

00:06:36.390 --> 00:06:38.355
Which is 4r squared.

00:06:42.220 --> 00:06:44.310
Area of the circle is
just pi r squared.

00:06:44.310 --> 00:06:46.530
You hopefully learned the
formula for area of a circle.

00:06:53.270 --> 00:06:55.940
Divide the numerator and the
denominator by r squared.

00:06:55.940 --> 00:06:57.760
You're left with pi/4.

00:06:57.760 --> 00:06:59.060
That's choice D.

00:07:01.570 --> 00:07:03.330
Problem 75.

00:07:03.330 --> 00:07:09.170
In the circle below, AB and CD
are chords intersecting at E.

00:07:09.170 --> 00:07:10.950
Fair enough.

00:07:10.950 --> 00:07:26.620
If AE is equal to 5, BE is
equal to 12, what is

00:07:26.620 --> 00:07:31.130
the value of DE?

00:07:31.130 --> 00:07:32.380
CE is equal to 6.

00:07:36.970 --> 00:07:39.290
What is the value of DE.

00:07:39.290 --> 00:07:41.260
Let's call that x.

00:07:41.260 --> 00:07:43.950
Now, I'm not going to prove it
here, just for saving time.

00:07:43.950 --> 00:07:47.470
But there's a neat property
of chords within a circle.

00:07:47.470 --> 00:07:49.970
That if I have two chords
intersecting a circle, it

00:07:49.970 --> 00:07:52.970
turns out that the two segments
when you multiply

00:07:52.970 --> 00:07:55.720
them times each other, are
always going to be equal to

00:07:55.720 --> 00:07:56.790
the same thing.

00:07:56.790 --> 00:08:03.560
So in this case, 5 times 12.

00:08:03.560 --> 00:08:10.390
So the two segments of chord
AB, so 5 times 12.

00:08:10.390 --> 00:08:13.510
That's going to be equal to
these two segments multiplied

00:08:13.510 --> 00:08:14.760
by each other.

00:08:17.870 --> 00:08:20.290
It's going to be equal
to x times 6.

00:08:20.290 --> 00:08:22.900
So you get 60 is equal to 6x.

00:08:22.900 --> 00:08:27.350
Divide both sides by 6, you
get x is equal to 10.

00:08:27.350 --> 00:08:31.700
And that is choice C.

00:08:31.700 --> 00:08:33.530
That might be a fun thing for
you to think about after this

00:08:33.530 --> 00:08:34.600
video of why that is.

00:08:34.600 --> 00:08:36.720
And maybe you want to play
around with chords and prove

00:08:36.720 --> 00:08:38.080
to yourself that that's
always the case.

00:08:38.080 --> 00:08:39.289
At least that it
makes intuition

00:08:39.289 --> 00:08:42.250
for you, makes sense.

00:08:42.250 --> 00:08:46.280
RB is tangent to a circle.

00:08:46.280 --> 00:08:49.910
Tangent means that it just
touches the outside of the

00:08:49.910 --> 00:08:51.810
circle right there at
only one point.

00:08:51.810 --> 00:08:54.300
And it's actually perpendicular
to the radius at

00:08:54.300 --> 00:08:55.870
that point.

00:08:55.870 --> 00:08:57.510
So this is the radius
of that point.

00:08:57.510 --> 00:08:58.920
The center is at A.

00:08:58.920 --> 00:09:00.100
This is a radius.

00:09:00.100 --> 00:09:02.740
And it's tangent at point B, so
it's perpendicular to the

00:09:02.740 --> 00:09:05.260
radius at that point.

00:09:05.260 --> 00:09:08.240
BD is a diameter,
OK, fair enough.

00:09:08.240 --> 00:09:11.775
Well A is the center, so
that's kind of obvious.

00:09:14.860 --> 00:09:17.070
So they want to know what is
the measure of angle CBR.

00:09:24.650 --> 00:09:29.040
So they want to know what
this angle is equal to.

00:09:29.040 --> 00:09:31.360
Well, I kind of did
it inadvertently.

00:09:31.360 --> 00:09:34.310
We know that when a line is
tangent to a circle, it's

00:09:34.310 --> 00:09:36.900
perpendicular to the radius
at that point.

00:09:36.900 --> 00:09:39.790
So this whole angle
is 90 degrees.

00:09:39.790 --> 00:09:41.330
So the angle that we're
trying to figure out,

00:09:41.330 --> 00:09:42.410
let's call that x.

00:09:42.410 --> 00:09:45.030
That's the complement to 25.

00:09:45.030 --> 00:09:48.460
x plus 25 is equal to 90.

00:09:48.460 --> 00:09:56.750
Subtract 25 from both sides.
x is equal to 65 degrees.

00:09:56.750 --> 00:09:58.545
And that is choice B.

00:10:02.550 --> 00:10:04.770
OK, I'll see you in
the next video.


CA Geometry: Basic trigonometry

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:02.850
We're on problem 61.

00:00:02.850 --> 00:00:07.170
It says the point minus 3,
2 lies on a circle whose

00:00:07.170 --> 00:00:11.990
equation is x plus 3 squared
plus y plus 1 squared is equal

00:00:11.990 --> 00:00:13.270
to r squared.

00:00:13.270 --> 00:00:18.080
Which of the following must be
the radius of the circle?

00:00:18.080 --> 00:00:21.100
So the way to think about it
is, is that this point

00:00:21.100 --> 00:00:22.770
satisfies this equation.

00:00:27.270 --> 00:00:30.800
Any point on the equation will
satisfy both sides of this

00:00:30.800 --> 00:00:31.500
equality sign.

00:00:31.500 --> 00:00:33.290
So all we have to do is
substitute the x and the y

00:00:33.290 --> 00:00:35.850
here and see what r squared
has to be equal to.

00:00:35.850 --> 00:00:38.110
So let's do that.

00:00:38.110 --> 00:00:40.080
If we substitute the
minus 3 in for x.

00:00:44.310 --> 00:00:48.100
You get minus 3, I just
substituted for the x.

00:00:48.100 --> 00:00:55.280
Plus 3 squared plus,
now y, y is 2.

00:00:55.280 --> 00:01:01.370
2 plus 1 squared is equal
to r squared.

00:01:01.370 --> 00:01:03.120
Minus 3 plus 3, that's just 0.

00:01:03.120 --> 00:01:04.410
0 squared is 0.

00:01:04.410 --> 00:01:06.180
And then 2 plus 1 squared.

00:01:06.180 --> 00:01:09.040
So 3 squared is equal
to r squared.

00:01:09.040 --> 00:01:11.080
You could say r squared is equal
to 9 and then r is equal

00:01:11.080 --> 00:01:14.060
to 3 because you can't have
a negative radius.

00:01:14.060 --> 00:01:16.925
We see immediately that
r is equal to 3.

00:01:16.925 --> 00:01:19.020
So all you have to
do is substitute

00:01:19.020 --> 00:01:20.920
the x and the y values.

00:01:20.920 --> 00:01:25.410
Because any point that satisfies
this equality is on

00:01:25.410 --> 00:01:27.490
the circle, defined
by this equation.

00:01:27.490 --> 00:01:29.460
They say this point is on the
circle, so you just have to

00:01:29.460 --> 00:01:32.620
substitute them in and
just solve for r.

00:01:32.620 --> 00:01:35.390
Problem 62.

00:01:35.390 --> 00:01:37.380
Looks like we're going to
do some trigonometry.

00:01:37.380 --> 00:01:41.860
In the figure, below if sine
of x is equal to 5 over 13

00:01:41.860 --> 00:01:44.570
what are the cosine of x
and the tangent of x?

00:01:44.570 --> 00:01:46.370
And I don't know if you've seen
the basic trigonometry

00:01:46.370 --> 00:01:48.190
videos, you might want to.

00:01:48.190 --> 00:01:51.450
But a good mnemonic for
memorizing sine, cosine and

00:01:51.450 --> 00:01:52.700
tangent is SOHCAHTOA.

00:01:59.380 --> 00:02:05.630
And that means SOH is sine is
equal to to opposite over

00:02:05.630 --> 00:02:06.970
hypotenuse.

00:02:06.970 --> 00:02:09.770
Cosine is equal to adjacent
over hypotenuse.

00:02:09.770 --> 00:02:11.750
And I'll tell you what these
mean in a second.

00:02:11.750 --> 00:02:15.760
And tangent, you might have
guessed, is equal to opposite

00:02:15.760 --> 00:02:17.250
over adjacent.

00:02:17.250 --> 00:02:18.270
So what does that mean?

00:02:18.270 --> 00:02:19.590
What is all of this mnemonic?

00:02:19.590 --> 00:02:20.810
So just you might want to
remember SOHCAHTOA.

00:02:20.810 --> 00:02:23.550
Then you could break
it down like that.

00:02:23.550 --> 00:02:26.550
So if I took the sine
of this angle.

00:02:26.550 --> 00:02:30.350
That means the opposite side
of this angle over the

00:02:30.350 --> 00:02:32.860
hypotenuse is equal to the
sine of this angle.

00:02:35.920 --> 00:02:37.170
Let's call this the opposite.

00:02:39.740 --> 00:02:42.530
This is the hypotenuse.

00:02:42.530 --> 00:02:44.310
This is the adjacent side.

00:02:44.310 --> 00:02:46.650
Because it's adjacent
to the angle.

00:02:46.650 --> 00:02:49.660
This one is opposite,
hypotenuse, adjacent.

00:02:49.660 --> 00:02:56.080
So the sine of x is equal to,
we know from our mnemonic

00:02:56.080 --> 00:03:01.760
SOHCAHTOA, opposite
over hypotenuse.

00:03:01.760 --> 00:03:10.300
And they tell us that that
is equal to 5/13.

00:03:10.300 --> 00:03:16.610
So opposite over hypotenuse
is equal to 5/13.

00:03:16.610 --> 00:03:18.970
Now we know that that's just
the ratio between the two.

00:03:18.970 --> 00:03:19.700
So we don't know.

00:03:19.700 --> 00:03:21.990
This could be 10, this
could be 26.

00:03:21.990 --> 00:03:25.010
This could be 1 and this
could be 13/5.

00:03:25.010 --> 00:03:25.730
Who knows.

00:03:25.730 --> 00:03:27.040
That actually doesn't matter.

00:03:27.040 --> 00:03:28.670
That's what's neat about
trigonometry.

00:03:28.670 --> 00:03:29.585
It's all about the ratio.

00:03:29.585 --> 00:03:32.160
So let's just assume
that this is 5.

00:03:32.160 --> 00:03:34.050
That the opposite
is equal to 5.

00:03:34.050 --> 00:03:36.430
And the hypotenuse
is equal to 13.

00:03:36.430 --> 00:03:37.960
Let me pick a different color.

00:03:37.960 --> 00:03:40.390
This is a little nauseating.

00:03:40.390 --> 00:03:40.860
All right.

00:03:40.860 --> 00:03:43.120
So if the opposite soon. is 5
and the hypotenuse is 13, what

00:03:43.120 --> 00:03:44.590
would the adjacent
be equal to?

00:03:44.590 --> 00:03:46.910
We could use the Pythagorean
theorem.

00:03:46.910 --> 00:03:49.700
So we could say the
adjacent squared.

00:03:49.700 --> 00:03:51.620
A squared plus the
opposite squared.

00:03:51.620 --> 00:03:54.850
So plus 5 squared, plus 25.

00:03:54.850 --> 00:03:56.940
Is equal to 13 squared.

00:03:56.940 --> 00:04:02.090
13 squared is 169.

00:04:02.090 --> 00:04:04.870
If you subtract 25 from both
sides of this equation, you

00:04:04.870 --> 00:04:08.900
get a squared is equal to 144.

00:04:08.900 --> 00:04:14.320
A is equal to 12.

00:04:14.320 --> 00:04:16.040
We don't know that a is
definitely equal to 12.

00:04:16.040 --> 00:04:18.920
But we know that the ratio
of the opposite to

00:04:18.920 --> 00:04:20.579
adjacent is 5 to 12.

00:04:20.579 --> 00:04:22.710
Because we just assumed that
the opposite is 5.

00:04:22.710 --> 00:04:25.890
Anyway, so they want to know
what are cosine of x and

00:04:25.890 --> 00:04:27.040
tangent of x.

00:04:27.040 --> 00:04:27.960
So CAH.

00:04:27.960 --> 00:04:29.520
SOHCAHTOA.

00:04:29.520 --> 00:04:38.770
Cosine of x is equal to the
adjacent over the hypotenuse.

00:04:38.770 --> 00:04:40.500
The adjacent is 12.

00:04:40.500 --> 00:04:41.710
Hypotenuse is 13.

00:04:41.710 --> 00:04:44.720
So it's equal to 12/13.

00:04:44.720 --> 00:04:46.460
That's the cosine of x.

00:04:46.460 --> 00:04:52.140
And the tangent of x is equal
to opposite over adjacent.

00:04:52.140 --> 00:04:53.590
TOA, opposite over adjacent.

00:04:53.590 --> 00:04:56.430
So opposite is 4,
adjacent is 12.

00:04:56.430 --> 00:04:58.080
Equal to 5/12.

00:04:58.080 --> 00:04:59.290
And let's see what
choice that is.

00:04:59.290 --> 00:05:02.110
That's choice A, cosine
of x is 12/13.

00:05:02.110 --> 00:05:05.550
Tangent of x is 5/12.

00:05:05.550 --> 00:05:08.530
Next question.

00:05:08.530 --> 00:05:10.140
Looks like they want us to learn
a lot of trigonometry

00:05:10.140 --> 00:05:10.760
and geometry.

00:05:10.760 --> 00:05:11.540
Which is good.

00:05:11.540 --> 00:05:14.040
This is getting you warmed
up for the trig.

00:05:14.040 --> 00:05:17.810
In the figure below, sine
of A is equal to 0.7.

00:05:17.810 --> 00:05:19.520
So let's call this angle a.

00:05:22.750 --> 00:05:24.680
They say what is the
length of AC?

00:05:24.680 --> 00:05:26.480
So we want to know that.

00:05:26.480 --> 00:05:28.970
Let's call that x.

00:05:28.970 --> 00:05:30.330
So SOHCAHTOA.

00:05:30.330 --> 00:05:39.410
SOH tells us that sine of some
angle, let's call that theta,

00:05:39.410 --> 00:05:43.220
is equal to the opposite
over the hypotenuse.

00:05:43.220 --> 00:05:52.090
So sine of A, in this example,
is going to be equal to the

00:05:52.090 --> 00:05:59.930
opposite, 21, over the
hypotenuse, over x.

00:05:59.930 --> 00:06:01.710
And they tell us that the sine
of A is equal to 0.7.

00:06:05.500 --> 00:06:09.080
So now we can just solve this
equation for x and we're done.

00:06:09.080 --> 00:06:10.410
Let's see.

00:06:10.410 --> 00:06:14.980
So if you multiply x times
both sides, you get 21 is

00:06:14.980 --> 00:06:17.670
equal to 0.7x.

00:06:17.670 --> 00:06:20.240
And you divide both
sides by 0.7.

00:06:20.240 --> 00:06:25.000
You get 21/0.7 is equal to x.

00:06:25.000 --> 00:06:26.510
21 divided by 7 is 3.

00:06:26.510 --> 00:06:30.740
So 21 divided by 0.7 is 30.

00:06:30.740 --> 00:06:36.730
So x is equal to 30.

00:06:36.730 --> 00:06:38.540
And that's length AC.

00:06:41.060 --> 00:06:43.170
That's choice C.

00:06:43.170 --> 00:06:44.030
Next question.

00:06:44.030 --> 00:06:46.360
64.

00:06:46.360 --> 00:06:50.910
Approximately how many feet
tall is the street light?

00:06:50.910 --> 00:06:54.580
OK, so we can use some
trigonometry here.

00:06:54.580 --> 00:06:58.440
So if we know this angle, and
they give us the all of the

00:06:58.440 --> 00:07:01.940
trig ratios for that angle,
we're trying to

00:07:01.940 --> 00:07:03.130
figure out the height.

00:07:03.130 --> 00:07:10.700
So if I write SOHCAHTOA, what
are we trying to figure?

00:07:10.700 --> 00:07:12.360
So they gave us the adjacent.

00:07:12.360 --> 00:07:16.110
This is adjacent to the angle,
it's right beside it.

00:07:16.110 --> 00:07:17.250
The height that we're
trying to figure

00:07:17.250 --> 00:07:18.500
out, this is the opposite.

00:07:20.810 --> 00:07:22.620
So if we can have a trig you
function that deals with the

00:07:22.620 --> 00:07:24.470
opposite and the adjacent.

00:07:24.470 --> 00:07:26.310
Well that's tangent.

00:07:26.310 --> 00:07:27.120
TOA.

00:07:27.120 --> 00:07:32.020
Tangent of any angle
is equal to the

00:07:32.020 --> 00:07:33.660
opposite over the adjacnet.

00:07:33.660 --> 00:07:42.030
In this case, tangent of 40
degrees is going to be equal

00:07:42.030 --> 00:07:45.940
to the opposite, the opposite is
h, that's what we're trying

00:07:45.940 --> 00:07:48.080
to solve for, over
the adjacent.

00:07:48.080 --> 00:07:50.780
The adjacent is 20 feet.

00:07:50.780 --> 00:07:54.120
OK, tangent of 40 degrees isn't
something that most

00:07:54.120 --> 00:07:58.330
people have memorized, that's OK
because they gave it to us.

00:07:58.330 --> 00:08:02.140
Tangent of 40 degrees is 0.84.

00:08:02.140 --> 00:08:07.850
So we get 0.84 is
equal to h/20.

00:08:07.850 --> 00:08:15.760
So we can multiply both sides of
that by 20 and we get h is

00:08:15.760 --> 00:08:21.710
equal to 20 times 0.84.

00:08:21.710 --> 00:08:39.490
And that is equal to 16.8.

00:08:39.490 --> 00:08:40.740
And that is choice C.

00:08:43.970 --> 00:08:45.900
Problem 65.

00:08:45.900 --> 00:08:50.570
Right triangle ABC is
pictured below.

00:08:50.570 --> 00:08:56.320
Which equation gives the
correct value for BC?

00:08:56.320 --> 00:08:59.200
So this is what they want
us to figure out.

00:08:59.200 --> 00:09:03.100
This is BC right there.

00:09:03.100 --> 00:09:05.970
OK, let's read them.

00:09:05.970 --> 00:09:08.710
Let me write SOHCAHTOA, I
actually do this a lot.

00:09:12.300 --> 00:09:18.070
OK, so they're saying that the
sine of 32 degrees is equal to

00:09:18.070 --> 00:09:19.230
BC over 8.2.

00:09:19.230 --> 00:09:19.970
Is that right?

00:09:19.970 --> 00:09:23.290
Sine is opposite over
hypotenuse.

00:09:23.290 --> 00:09:26.550
BC is definitely the opposite.

00:09:26.550 --> 00:09:29.760
8.2 is not the hypotenuse,
10.6 is the hypotenuse.

00:09:29.760 --> 00:09:31.780
So they're doing, this
is the adjacent.

00:09:31.780 --> 00:09:33.030
So this is wrong.

00:09:36.230 --> 00:09:37.890
So this should be a tangent.

00:09:37.890 --> 00:09:41.590
Tangent of 32 is equal
to BC over 8.2.

00:09:41.590 --> 00:09:45.740
This is the adjacent side,
adjacent to 32 degrees.

00:09:45.740 --> 00:09:47.580
That's the opposite, and
that's the hypotenuse.

00:09:47.580 --> 00:09:49.750
So that's not right.

00:09:49.750 --> 00:09:51.190
Choice B, cosine of 32.

00:09:51.190 --> 00:09:54.160
Cosine is adjacent
over hypotenuse.

00:09:54.160 --> 00:10:00.210
So cosine of 32 should
be 8.2/10.6.

00:10:00.210 --> 00:10:02.870
So this should be an 8.2 here.

00:10:02.870 --> 00:10:04.980
So this isn't right.

00:10:04.980 --> 00:10:08.550
OK, so the next one, tangent
of 58 degrees.

00:10:08.550 --> 00:10:10.050
Where are they getting
that 58?

00:10:10.050 --> 00:10:12.940
Well, they know that this is
32, this is 90, so this is

00:10:12.940 --> 00:10:15.600
going to be 180 minus
32 minus 90.

00:10:15.600 --> 00:10:18.300
Because the angles in a triangle
add up to 180.

00:10:18.300 --> 00:10:23.670
So this angle right
here is 58.

00:10:23.670 --> 00:10:26.390
And now if we use that angle,
we have to relabel opposite

00:10:26.390 --> 00:10:28.190
and adjacent and all that.

00:10:28.190 --> 00:10:32.770
So from this angle's point
of view, tangent is

00:10:32.770 --> 00:10:35.420
opposite over adjacent.

00:10:35.420 --> 00:10:40.040
So if we write the tangent of
58 is equal to the opposite

00:10:40.040 --> 00:10:44.820
side, should be equal
to 8.2 over its

00:10:44.820 --> 00:10:47.800
adjacent side, over BC.

00:10:47.800 --> 00:10:49.190
This is adjacent
to this angle.

00:10:49.190 --> 00:10:51.080
It was opposite this
angle, but BC is

00:10:51.080 --> 00:10:52.760
adjacent to this angle.

00:10:52.760 --> 00:10:54.040
So that's what they wrote.

00:10:54.040 --> 00:10:57.770
So choice C is correct.

00:10:57.770 --> 00:10:58.630
And we're done.

00:10:58.630 --> 00:10:59.850
I'll see you in the
next video.

00:10:59.850 --> 00:11:01.180
Well we're not done with
the whole thing.

00:11:01.180 --> 00:11:02.180
I'm done with this video.

00:11:02.180 --> 00:11:03.430
See


CA Geometry: More trig

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.670
We're on problem 66.

00:00:02.670 --> 00:00:04.550
In the accompanying diagram,
the measure of

00:00:04.550 --> 00:00:06.540
angle a is 32 degrees.

00:00:06.540 --> 00:00:08.770
And AC is equal to 10.

00:00:08.770 --> 00:00:13.060
Which equation could be used
to find x in triangle ABC?

00:00:13.060 --> 00:00:14.380
So we want x.

00:00:14.380 --> 00:00:15.960
So let's write our
SOHCAHTOA down.

00:00:19.250 --> 00:00:20.880
What are they giving us?

00:00:20.880 --> 00:00:24.250
Well x is the opposite side,
this is equal to the opposite,

00:00:24.250 --> 00:00:26.760
it's opposite of our
angle in question.

00:00:26.760 --> 00:00:29.190
And then this 10 side
is the adjacent.

00:00:29.190 --> 00:00:32.150
So I haven't looked at the
choices yet, but if we say the

00:00:32.150 --> 00:00:40.650
tangent of 32 degrees should be
equal to the opposite side,

00:00:40.650 --> 00:00:45.670
which is x, over the adjacent
side, which is 10.

00:00:45.670 --> 00:00:52.870
And then if we multiply both
sides of this by 10, we get 10

00:00:52.870 --> 00:00:59.580
tangent of 32 degrees
is equal to x.

00:00:59.580 --> 00:01:01.860
Let's hope that's one
of the choices.

00:01:01.860 --> 00:01:04.410
Yep, sure enough, C.

00:01:04.410 --> 00:01:07.270
x is equal to 10 tangent
of 32 degrees.

00:01:07.270 --> 00:01:09.200
Everything in trigonometry
really does

00:01:09.200 --> 00:01:10.310
boil down to SOHCAHTOA.

00:01:10.310 --> 00:01:11.350
You'll be amazed.

00:01:11.350 --> 00:01:13.910
When you actually take
trigonometry as a class,

00:01:13.910 --> 00:01:15.420
you'll see that they
have all these trig

00:01:15.420 --> 00:01:16.350
identities and all that.

00:01:16.350 --> 00:01:20.160
But all of those things can be
just proved from SOHCAHTOA.

00:01:20.160 --> 00:01:22.880
And these are just definitions
of ratios that have been very

00:01:22.880 --> 00:01:26.280
useful in the world when
studying triangles.

00:01:26.280 --> 00:01:27.700
All right, 67.

00:01:27.700 --> 00:01:30.010
But you can watch, we have a
whole playlist on trigonometry

00:01:30.010 --> 00:01:32.610
and we talk about all the
ratios and the unit

00:01:32.610 --> 00:01:33.320
circle and all that.

00:01:33.320 --> 00:01:35.500
So if you're enjoying these
problems, I encourage you to

00:01:35.500 --> 00:01:36.520
watch those videos.

00:01:36.520 --> 00:01:38.455
Or if you don't understand these
problems, I encourage

00:01:38.455 --> 00:01:40.870
you even more to watch
those videos.

00:01:40.870 --> 00:01:42.240
All right.

00:01:42.240 --> 00:01:46.610
The diagram shows an 8 foot
ladder leaning against a wall.

00:01:46.610 --> 00:01:49.800
The ladder makes a 53 degree
angle with the wall.

00:01:49.800 --> 00:01:53.030
Which is closest to the distance
up the wall the

00:01:53.030 --> 00:01:54.230
ladder reaches?

00:01:54.230 --> 00:01:55.990
So this is what they
care about, this

00:01:55.990 --> 00:01:58.810
distance right here.

00:01:58.810 --> 00:02:02.430
So let's see how we
can figure it out.

00:02:02.430 --> 00:02:05.270
If we write SOHCAHTOA, maybe
we don't have to

00:02:05.270 --> 00:02:06.620
to write that anymore.

00:02:06.620 --> 00:02:10.460
So we have this angle and
what have they given us?

00:02:10.460 --> 00:02:12.910
This deals with the hypotenuse,
which is 8.

00:02:12.910 --> 00:02:14.535
And it deals with this
adjacent side.

00:02:18.290 --> 00:02:21.630
So what trig function deals with
the adjacent side and the

00:02:21.630 --> 00:02:23.020
hypotenuse.

00:02:23.020 --> 00:02:23.820
Well, I'll write it down.

00:02:23.820 --> 00:02:25.070
SOHCAHTOA.

00:02:28.660 --> 00:02:30.050
We want to figure out adjacent,
we have the

00:02:30.050 --> 00:02:30.540
hypotenuse.

00:02:30.540 --> 00:02:31.940
What trig function
deals with it?

00:02:31.940 --> 00:02:35.100
Well, cosine is adjacent
over hypotenuse.

00:02:35.100 --> 00:02:42.870
So the cosine of 53 degrees is
equal to the adjacent side of

00:02:42.870 --> 00:02:44.950
this triangle, of this
right triangle.

00:02:44.950 --> 00:02:47.190
So that's A.

00:02:47.190 --> 00:02:49.960
And you should only be applying
these, right now,

00:02:49.960 --> 00:02:51.220
with right triangles.

00:02:51.220 --> 00:02:53.610
Later on we'll learn, in
trigonometry how trig

00:02:53.610 --> 00:02:56.640
functions are useful
for any triangle.

00:02:56.640 --> 00:02:58.480
Because obviously, only
a right triangle has a

00:02:58.480 --> 00:02:59.050
hypotenuse.

00:02:59.050 --> 00:03:02.330
So these h's wouldn't be too
meaningful if you're not

00:03:02.330 --> 00:03:03.470
dealing with a right triangle.

00:03:03.470 --> 00:03:04.380
Back to the problem.

00:03:04.380 --> 00:03:07.520
Cosine of 53, it equals the
adjacent side over the

00:03:07.520 --> 00:03:11.410
hypotenuse, which is 8.

00:03:11.410 --> 00:03:14.960
I can write x there, x
is the adjacent side.

00:03:14.960 --> 00:03:18.720
So we multiply both sides of
this by 8, we get 8 times the

00:03:18.720 --> 00:03:23.460
cosine of 53 degrees
is equal to x.

00:03:23.460 --> 00:03:25.960
And they give us
a little chart.

00:03:25.960 --> 00:03:28.680
They tell us that the cosine of
53 degrees is approximately

00:03:28.680 --> 00:03:31.240
equal to 0.6.

00:03:31.240 --> 00:03:36.070
So we get 8 times 0.6
is equal to x.

00:03:36.070 --> 00:03:37.770
8 times 0.6 is 4.8.

00:03:41.360 --> 00:03:44.800
So x is equal to 4.8.

00:03:44.800 --> 00:03:47.420
And that is choice B.

00:03:51.170 --> 00:03:52.920
Problem 68.

00:03:52.920 --> 00:03:56.640
Triangle JKL is shown below.

00:03:56.640 --> 00:03:57.200
Fair enough.

00:03:57.200 --> 00:04:02.530
Which equation should be used
to find the length of JK?

00:04:02.530 --> 00:04:06.560
This is JK.

00:04:06.560 --> 00:04:11.270
All right so we have
this angle here.

00:04:11.270 --> 00:04:14.920
JK is the opposite side.

00:04:14.920 --> 00:04:16.480
We want to figure out
the opposite side.

00:04:16.480 --> 00:04:17.519
We have the hypotenuse.

00:04:17.519 --> 00:04:20.920
So what trig function deals with
opposite and hypotenuse?

00:04:20.920 --> 00:04:22.800
Well I could just
write SOH down.

00:04:22.800 --> 00:04:24.970
I'm don't even have to write
the whole SOHCAHTOA.

00:04:24.970 --> 00:04:28.210
Because sine is equal to the
opposite side over the

00:04:28.210 --> 00:04:29.350
hypotenuse.

00:04:29.350 --> 00:04:36.520
So in this case, sine of 24
degrees is equal to the

00:04:36.520 --> 00:04:40.780
opposite side, which is side
JK, over the hypotenuse.

00:04:40.780 --> 00:04:42.840
This is the hypotenuse
right here, 28.

00:04:42.840 --> 00:04:44.840
The longest side of the right
triangle, or the side that's

00:04:44.840 --> 00:04:46.800
opposite the 90 degree angle.

00:04:46.800 --> 00:04:50.160
And so maybe we're done.

00:04:50.160 --> 00:04:59.690
So sine of 24 degrees is
equal to JK over 28.

00:04:59.690 --> 00:05:00.940
That's choice A.

00:05:08.380 --> 00:05:11.070
Problem 69.

00:05:11.070 --> 00:05:13.650
These are kind of fun,
aren't they?

00:05:13.650 --> 00:05:16.710
What is the approximate height
in feet of the tree in the

00:05:16.710 --> 00:05:17.260
figure below?

00:05:17.260 --> 00:05:19.080
Now you'll see why trigonometry
is useful,

00:05:19.080 --> 00:05:22.160
because I'm sure you often
wonder how tall is that tree?

00:05:22.160 --> 00:05:25.050
And it might be hard
to climb it.

00:05:25.050 --> 00:05:26.320
And now you can use
trigonometry.

00:05:26.320 --> 00:05:28.440
If you just have a way to
measure the angle between your

00:05:28.440 --> 00:05:35.350
line of sight and between you
and the top of the tree.

00:05:35.350 --> 00:05:39.120
You've probably seen those
surveyors, actually they use

00:05:39.120 --> 00:05:39.760
things like this.

00:05:39.760 --> 00:05:42.090
And you can figure out the
heights of things far away if

00:05:42.090 --> 00:05:43.630
you know how far away
you are from it.

00:05:43.630 --> 00:05:46.070
But anyway, let's work
on this problem.

00:05:46.070 --> 00:05:48.880
What is the approximate height
in feet of the tree in the

00:05:48.880 --> 00:05:50.580
figure below?

00:05:50.580 --> 00:05:53.990
OK, so this is 50.

00:05:53.990 --> 00:05:55.580
So let's think about it.

00:05:55.580 --> 00:05:56.780
What do they tell us?

00:05:56.780 --> 00:05:59.460
This is 90, this is 50.

00:05:59.460 --> 00:06:02.390
So the approximate
height in feet.

00:06:02.390 --> 00:06:07.120
OK, so this is the opposite
side of this angle.

00:06:07.120 --> 00:06:08.320
So let's call this h.

00:06:08.320 --> 00:06:11.480
So what deals with the
opposite side?

00:06:11.480 --> 00:06:14.125
Actually I should use h,
let's call that O.

00:06:14.125 --> 00:06:15.450
O is the height of the
tree because it's

00:06:15.450 --> 00:06:17.930
opposite the 50 angle.

00:06:17.930 --> 00:06:20.870
So what deals with the opposite
and the hypotenuse?

00:06:20.870 --> 00:06:23.580
So once again, SOHCAHTOA.

00:06:23.580 --> 00:06:25.220
We just have to look
at the SOHA part.

00:06:25.220 --> 00:06:27.870
Sine is equal to opposite
over hypotenuse.

00:06:27.870 --> 00:06:36.790
So the sine of 50 degrees is
equal to the opposite side of

00:06:36.790 --> 00:06:39.660
this triangle, that's the
height of the tree.

00:06:39.660 --> 00:06:42.810
So I'll say O for the
height of the tree.

00:06:42.810 --> 00:06:46.010
Over the hypotenuse, the
hypotenuse is 100 feet.

00:06:46.010 --> 00:06:47.730
Multiply both sides by 100.

00:06:47.730 --> 00:06:53.710
You get 100 sine of 50 is equal
to this opposite side,

00:06:53.710 --> 00:06:55.430
which is equal to the
height of the tree.

00:06:55.430 --> 00:06:59.830
Sine of 50 is 0.76.

00:06:59.830 --> 00:07:04.360
So 0.76 times 100
is equal to O.

00:07:04.360 --> 00:07:07.610
That 76 is equal to O, or
the height of the tree.

00:07:10.230 --> 00:07:11.600
Oh, they said 0.766.

00:07:11.600 --> 00:07:13.160
So let's put another 6.

00:07:13.160 --> 00:07:16.480
That becomes 76.6.

00:07:16.480 --> 00:07:19.730
And that's choice B.

00:07:19.730 --> 00:07:21.720
All right.

00:07:21.720 --> 00:07:22.560
Problem 70.

00:07:22.560 --> 00:07:27.320
These go fast. If a is equal
to 3 roots of 3 in the

00:07:27.320 --> 00:07:28.570
triangle below.

00:07:31.360 --> 00:07:33.690
I think you're going to learn a
little bit about 30, 60, 90

00:07:33.690 --> 00:07:35.140
triangles now.

00:07:35.140 --> 00:07:37.695
What is the value of B?

00:07:41.250 --> 00:07:43.600
OK, maybe they assume that you
already do know something

00:07:43.600 --> 00:07:46.070
about 30, 60, 90 triangles.

00:07:46.070 --> 00:07:49.330
But anyway, if a is equal
to 3 root 3, what is B?

00:07:49.330 --> 00:07:51.830
And I'm debating on what
I should assume

00:07:51.830 --> 00:07:54.130
that you already know.

00:07:54.130 --> 00:07:55.530
But let's think about
it a little bit.

00:07:58.040 --> 00:08:01.090
Let me see how I can do this
without resorting to just the

00:08:01.090 --> 00:08:06.200
definition of 30, 60,
90 triangles.

00:08:06.200 --> 00:08:07.350
I'll prove it for you.

00:08:07.350 --> 00:08:08.940
Although, later you
can memorize.

00:08:08.940 --> 00:08:11.360
Because when I keep saying 30,
60, 90 triangles, we know that

00:08:11.360 --> 00:08:14.260
this right here is 60 degrees.

00:08:14.260 --> 00:08:18.040
Because 30, plus 60, plus 90
is equal to 180 degrees.

00:08:18.040 --> 00:08:20.090
So that's why we could
we keep calling this

00:08:20.090 --> 00:08:22.510
a 30, 60, 90 triangle.

00:08:22.510 --> 00:08:25.280
So how do we figure out
this side, if we

00:08:25.280 --> 00:08:30.310
just know that side?

00:08:30.310 --> 00:08:31.830
And we know this
is 30 degrees.

00:08:31.830 --> 00:08:34.840
So what I'm going to do is,
I'm going to try to redraw

00:08:34.840 --> 00:08:37.620
this same triangle but I'm
going to flip it over.

00:08:37.620 --> 00:08:40.799
And this is actually the proof
of figuring out the measures

00:08:40.799 --> 00:08:42.950
of a 30, 60, 90 triangle.

00:08:42.950 --> 00:08:44.330
So maybe I'll do it in
a different color.

00:08:44.330 --> 00:08:47.420
And I have some time
so I'll do it.

00:08:47.420 --> 00:08:49.840
Let's see.

00:08:49.840 --> 00:08:52.520
So I'll do it like that.

00:08:52.520 --> 00:08:55.840
I have to bring another line
down, something like that.

00:08:55.840 --> 00:09:00.495
And then I'll have to draw
another line more like that.

00:09:00.495 --> 00:09:02.590
I think you get the idea.

00:09:02.590 --> 00:09:05.660
And then bring a line
across like that.

00:09:05.660 --> 00:09:07.930
And actually, I'm going to draw
another line straight

00:09:07.930 --> 00:09:09.360
down like that.

00:09:09.360 --> 00:09:11.360
OK, let's see what we
can do with this.

00:09:11.360 --> 00:09:12.660
Maybe there's an easier
way to do it.

00:09:12.660 --> 00:09:15.400
But this is just what my brain
is thinking of right now.

00:09:15.400 --> 00:09:17.140
Let's think about
it a little bit.

00:09:17.140 --> 00:09:21.510
In this drawing that I've done,
this triangle is just a

00:09:21.510 --> 00:09:23.590
mirror image of that one.

00:09:23.590 --> 00:09:28.060
So this right here,
this is also 60.

00:09:28.060 --> 00:09:30.990
This is 60, this is 60, what's
this angle going to be?

00:09:30.990 --> 00:09:33.710
Well they're all collectively
supplements of each other.

00:09:33.710 --> 00:09:35.030
If you go all the way
from there to there,

00:09:35.030 --> 00:09:36.340
you have 180 degrees.

00:09:36.340 --> 00:09:41.540
So this big angle right here
has got to be 60 degrees.

00:09:41.540 --> 00:09:44.340
This angle right here, it's
a complement of 30.

00:09:44.340 --> 00:09:46.140
So this is 60 degrees.

00:09:46.140 --> 00:09:49.890
And then this is 60 degrees.

00:09:49.890 --> 00:09:51.380
So this is an equilateral
triangle.

00:09:51.380 --> 00:09:52.550
All the sides are going
to be equal.

00:09:52.550 --> 00:09:56.690
So that side is equal to
that side, which is

00:09:56.690 --> 00:09:59.650
equal to this side.

00:09:59.650 --> 00:10:01.230
Now let me ask another
question.

00:10:01.230 --> 00:10:05.650
What is this side right here?

00:10:05.650 --> 00:10:08.630
Let me just draw the
shorter part.

00:10:08.630 --> 00:10:11.050
And once you memorize 30, 60,
90 triangles, you don't have

00:10:11.050 --> 00:10:12.180
to go through all this.

00:10:12.180 --> 00:10:14.310
But it's good to be able
to reprove it.

00:10:14.310 --> 00:10:16.460
So what's this length
right here?

00:10:16.460 --> 00:10:19.190
Well if we look right up
there, that's length a.

00:10:19.190 --> 00:10:23.750
This was just a mirror image,
so this is also length a.

00:10:23.750 --> 00:10:26.860
This whole base of the
equilateral triangle is 2a.

00:10:26.860 --> 00:10:28.700
Well it's an equilateral
triangle so all the sides are

00:10:28.700 --> 00:10:32.330
the same, so this is
2a and this is 2a.

00:10:32.330 --> 00:10:34.890
And just like that we were
able to figure out the

00:10:34.890 --> 00:10:36.290
hypotenuse.

00:10:36.290 --> 00:10:37.580
And they want to know
what the b is.

00:10:37.580 --> 00:10:39.660
Once we know two sides of a
right triangle, it's very easy

00:10:39.660 --> 00:10:41.230
to figure out the third side.

00:10:41.230 --> 00:10:45.270
So we know that a squared
plus b squared

00:10:45.270 --> 00:10:46.000
is equal to c squared.

00:10:46.000 --> 00:10:47.250
Let me write that down.

00:10:52.900 --> 00:10:55.036
Now what's a squared?

00:10:58.670 --> 00:11:00.510
a is 3 roots of 3.

00:11:00.510 --> 00:11:05.680
So a squared would be, let me
write it down, 3 roots of 3

00:11:05.680 --> 00:11:09.420
squared plus b squared is equal
to, let me do it in

00:11:09.420 --> 00:11:10.670
another color.

00:11:10.670 --> 00:11:14.320
What's c? c, we just figured
out, is 2 times a.

00:11:14.320 --> 00:11:19.630
So it's 6 roots of 3 squared.

00:11:19.630 --> 00:11:21.680
That's what we did all this
stuff for, to figure out that

00:11:21.680 --> 00:11:24.370
this length is twice
that length.

00:11:24.370 --> 00:11:25.500
Fair enough.

00:11:25.500 --> 00:11:26.340
Now let's simplify.

00:11:26.340 --> 00:11:29.990
So if we take 3 roots of 3 to
the second power, that's the

00:11:29.990 --> 00:11:34.060
same thing as 3 squared times
square root of 3 squared.

00:11:34.060 --> 00:11:39.450
So that's 9 times 3
plus b squared is

00:11:39.450 --> 00:11:42.920
equal to 36 times 3.

00:11:42.920 --> 00:11:47.820
And so that's 27
plus b squared.

00:11:47.820 --> 00:11:52.030
36 times 3 is 108.

00:11:52.030 --> 00:11:54.060
Subtract 27 from both sides.

00:11:54.060 --> 00:11:59.300
b squared is equal to 81.

00:11:59.300 --> 00:12:02.070
b is equal to 9.

00:12:02.070 --> 00:12:04.160
So that is choice a.

00:12:04.160 --> 00:12:06.550
Anyway, you should watch the
videos I've done in the

00:12:06.550 --> 00:12:09.410
trigonometry playlist on 30, 60,
90 triangles if you want

00:12:09.410 --> 00:12:10.570
to be able to do this faster.

00:12:10.570 --> 00:12:11.440
But I think that was useful.

00:12:11.440 --> 00:12:14.770
Because you've actually seem
how you can figure out the

00:12:14.770 --> 00:12:17.130
sides of a 30, 60, 90 triangle
without having memorized it

00:12:17.130 --> 00:12:17.810
ahead of time.

00:12:17.810 --> 00:12:20.170
Anyway, see you in
the next video.