WEBVTT mathematics/geometry/pyo
00:00:00.000 --> 00:00:01.700
Welcome back to Educator.com.
00:00:01.700 --> 00:00:06.500
We are going to go over the geometric mean for our next lesson.
00:00:06.500 --> 00:00:18.700
For the geometric mean, here, we have a/x, which is equal to x/b.
00:00:18.700 --> 00:00:25.100
Now, if you remember, a few lessons ago, we went over ratios and proportions.
00:00:25.100 --> 00:00:32.100
When we have a proportion, we have two ratios that are equal to each other.
00:00:32.100 --> 00:00:49.300
Here, these numbers right here are called the means, and these numbers here are called extremes.
00:00:49.300 --> 00:01:00.100
So, when we talk about the geometric mean, we are talking about this number and this number.
00:01:00.100 --> 00:01:06.700
The geometric mean between two positive numbers, a and b, is x.
00:01:06.700 --> 00:01:13.500
If we are given two numbers, a and b, those numbers would be considered the extremes.
00:01:13.500 --> 00:01:22.100
And then, x, which is the geometric mean, is going to go here and here.
00:01:22.100 --> 00:01:34.300
Now, when we solve proportions, remember: we cross-multiply; so we are going to do this number times this number, is equal to a times b.
00:01:34.300 --> 00:01:54.500
If we are asked to find the geometric mean between two positive numbers--let's say 3 and 8--find the geometric mean between 3 and 8...
00:01:54.500 --> 00:02:01.300
Now, actually, before we continue with that problem, let's go back to this.
00:02:01.300 --> 00:02:13.900
If we cross-multiply, then we are going to get x² = a times b; then how would you find x?
00:02:13.900 --> 00:02:20.200
x is going to be the square root of ab.
00:02:20.200 --> 00:02:28.500
The geometric mean x is going to be the square root of this number times that number.
00:02:28.500 --> 00:02:32.300
This is how you would find the geometric mean between the two numbers.
00:02:32.300 --> 00:02:39.400
Now, back to this: 3 and 8--find the geometric mean between 3 and 8.
00:02:39.400 --> 00:02:48.400
What you can do is go ahead and solve it like this: plug it into here and find the square root of 3 times 8.
00:02:48.400 --> 00:02:59.400
Or you can just first set it up as a proportion; and this actually makes it a lot easier to see, especially if you are given
00:02:59.400 --> 00:03:06.200
different versions of this problem, like if you are given the geometric mean and asked to find one of these numbers.
00:03:06.200 --> 00:03:15.800
It is always easiest to put it into this proportion first, because you know that these two numbers here have to be the geometric mean.
00:03:15.800 --> 00:03:29.000
If they are asking you to find the geometric mean, that means that that would be this number down here, and that number up there, for your proportion.
00:03:29.000 --> 00:03:36.400
And then, 3 and 8 are going to go here and here; so this is how you would set it up.
00:03:36.400 --> 00:03:40.400
Find the geometric mean--that means that these numbers right here are what you are looking for.
00:03:40.400 --> 00:03:55.400
To solve it out, cross-multiply: x² is equal to 3 times 8, which is 24, so x is equal to √24.
00:03:55.400 --> 00:04:06.900
And that is going to be 2√6, or you can use your calculator to change that to a decimal.
00:04:06.900 --> 00:04:14.000
But this would be the answer, right here: this is the geometric mean between 3 and 8.
00:04:14.000 --> 00:04:22.000
Again, set it up as a proportion; put the two numbers up here and down there.
00:04:22.000 --> 00:04:34.300
Make sure that the geometric mean goes here and here, because those are the means; and then, just cross-multiply to solve.
00:04:34.300 --> 00:04:46.200
Similar triangles: now, here we have a right triangle, and remember: an altitude is a segment
00:04:46.200 --> 00:04:59.000
starting from a vertex and going to the side opposite that vertex so that it is perpendicular to it.
00:04:59.000 --> 00:05:07.000
If we draw an altitude from this vertex (and it has to be drawn from that right angle)--
00:05:07.000 --> 00:05:17.800
so then here is our right angle, and you are going to draw an altitude so that it is perpendicular to the side opposite--
00:05:17.800 --> 00:05:24.900
in this case, it is the hypotenuse, because it is coming from that right angle--then the two triangles formed
00:05:24.900 --> 00:05:29.700
are similar to the given triangle and to each other.
00:05:29.700 --> 00:05:35.000
Before this altitude was drawn, we only had one triangle.
00:05:35.000 --> 00:05:45.000
Now, after the altitude, we have three triangles: we have the big one; we have this one right here; and we have this one right here--three triangles.
00:05:45.000 --> 00:05:55.700
Now, this theorem is saying that, once that altitude is drawn from that right angle to the hypotenuse, all three triangles are now similar.
00:05:55.700 --> 00:06:07.900
Remember: similar triangles are triangles that have congruent angles, but proportional sides.
00:06:07.900 --> 00:06:17.500
"Similar" just means that they have the same shape, but a different size; that is "similar" or "similarity."
00:06:17.500 --> 00:06:26.100
So, here we know that we have three triangles--the same shape, but slightly different sizes.
00:06:26.100 --> 00:06:37.400
To state all three triangles so that the corresponding parts are in order, so that we can say that they are similar,
00:06:37.400 --> 00:06:45.900
I can name all three; it doesn't matter which order, so if I want to say the big one first:
00:06:45.900 --> 00:06:59.000
the big one is going to be triangle ABC; it is similar to...and then we are going to just name another triangle.
00:06:59.000 --> 00:07:09.000
Now, remember: it has to be corresponding to the order; so AB in the big triangle
00:07:09.000 --> 00:07:25.500
(and that is kind of hard to see)...the easiest way to see their corresponding parts is to look at it from long leg
00:07:25.500 --> 00:07:32.800
to short leg, and then hypotenuse; those are the three parts that make up a right triangle.
00:07:32.800 --> 00:07:43.700
It is always easiest to determine whatever you are stating--which side it would be considered from that triangle.
00:07:43.700 --> 00:07:48.200
And then, you can just look at the corresponding part of the other triangle.
00:07:48.200 --> 00:07:59.300
So, AB from the big triangle is considered the long leg; this would be AB.
00:07:59.300 --> 00:08:07.500
AB is the long leg; BC is the short leg; and then, AC is the hypotenuse.
00:08:07.500 --> 00:08:11.100
If AB is the long leg, then I have to mention that one first.
00:08:11.100 --> 00:08:19.200
And remember: B is the right angle; so then, the right angle is going to go in the middle of the next triangle that I am stating.
00:08:19.200 --> 00:08:30.100
Then, let's say I am going to name this triangle--not the very small one, but the medium-sized one, this one right here.
00:08:30.100 --> 00:08:36.200
I am going to label this D; so remember, I have to state the long first.
00:08:36.200 --> 00:08:45.100
The long leg from this triangle would be AD, because here is the right angle; AD is the long leg.
00:08:45.100 --> 00:08:48.600
Now, does it matter if I say AD or DA?
00:08:48.600 --> 00:09:00.100
Well, I know that, since B is my right angle for the first triangle, I have to state the right angle in that same order.
00:09:00.100 --> 00:09:16.400
So, it is going to be AD, triangle AD, because AD is my long leg, and D is my right angle; and then, it would be AD, and then B.
00:09:16.400 --> 00:09:31.500
So again, from the big triangle, it is triangle ABC, with AB as a long leg, BC as a short leg, and AC as the hypotenuse.
00:09:31.500 --> 00:09:43.000
For my second triangle that I am naming, AD is my long leg; BD is my short leg; and AB is my hypotenuse.
00:09:43.000 --> 00:09:52.900
And then, see how D is the right angle; the middle angle that I am listing is the right angle.
00:09:52.900 --> 00:10:00.100
So then, the third triangle has to be in the same order, so from this one right here,
00:10:00.100 --> 00:10:05.900
which one is my long leg? Which one is my short leg? And which one is my hypotenuse?
00:10:05.900 --> 00:10:13.200
My long leg from here...this is the right angle...would be BD or DB.
00:10:13.200 --> 00:10:18.000
Now, from this triangle, where is my right angle?--this one, D.
00:10:18.000 --> 00:10:22.400
I know that that letter D is going to go in the middle.
00:10:22.400 --> 00:10:34.300
So then, BD is my long leg; D is my right angle; C is my missing vertex.
00:10:34.300 --> 00:10:42.000
BD is my long leg; DC is my short leg; and BC is my hypotenuse.
00:10:42.000 --> 00:10:53.200
So now, I have listed all three triangles within their corresponding parts.
00:10:53.200 --> 00:11:11.700
Just keep in mind that, when the altitude is drawn from the right angle, you are actually splitting this up into three similar triangles--not congruent, but similar.
00:11:11.700 --> 00:11:24.700
Here, we have a right triangle, again, with the altitude drawn from the right angle of that triangle down to the hypotenuse.
00:11:24.700 --> 00:11:50.200
When you have this diagram, the measure of the altitude, BD, is going to be the geometric mean between AD and DC.
00:11:50.200 --> 00:11:59.100
Remember how, when we went over geometric mean, we found the geometric mean between two numbers.
00:11:59.100 --> 00:12:10.900
So then, the two numbers that are given are a and b, and then, this is a geometric mean and a geometric mean.
00:12:10.900 --> 00:12:19.600
This right here, BD, would be the geometric mean between the two parts of the hypotenuse.
00:12:19.600 --> 00:12:28.100
Just to read this to you: the measures of the altitude drawn from the vertex of the right angle of the right triangle
00:12:28.100 --> 00:12:36.000
to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.
00:12:36.000 --> 00:12:45.100
So, because this right here--see how it splits up the hypotenuse into two parts, AD and DC--those two parts,
00:12:45.100 --> 00:12:58.300
then it is as if AD is a here; DC is b; so then, those are the two numbers; and BD, the altitude, would be the geometric mean.
00:12:58.300 --> 00:13:08.300
So then, if you were to write this out, this would be like BD is the geometric mean between AD and DC, these two parts.
00:13:08.300 --> 00:13:18.100
Then, AD would go up here, and DC would go right there; that is like a and b--those are the two numbers.
00:13:18.100 --> 00:13:28.600
And then, the geometric mean, which is the altitude, BD, is going to go here and here.
00:13:28.600 --> 00:13:35.900
So, whenever they say that something is the geometric mean, that something goes in here and here,
00:13:35.900 --> 00:13:43.600
because remember, again: these two would be the means.
00:13:43.600 --> 00:13:58.300
Let's say that this is x, and we want to find the geometric mean between...let's say AD is 20 and DC is 5.
00:13:58.300 --> 00:14:00.400
So then, we are trying to find the altitude.
00:14:00.400 --> 00:14:15.400
Now, we know, because of this theorem, that the altitude, BD, is the geometric mean between AD and DC, these two parts.
00:14:15.400 --> 00:14:23.600
And this would be the geometric mean; so to solve it out, I am going to make it into my proportion.
00:14:23.600 --> 00:14:27.300
The geometric mean is BD; that is going to go here and here.
00:14:27.300 --> 00:14:39.300
My two numbers, a and b, are going to go here and here; so when I solve it out, it is going to become x² = 100, so x is 10.
00:14:39.300 --> 00:14:48.100
So then, this right here...the measure of the altitude is 10.
00:14:48.100 --> 00:14:57.000
Now, one more: here, for this one, if the altitude is drawn to the hypotenuse of a right triangle,
00:14:57.000 --> 00:15:09.500
then the measure of a leg of the triangle (we have two legs: it is AB and BC, the two legs of the big triangle)
00:15:09.500 --> 00:15:16.700
is the geometric mean between the measures of the hypotenuse
00:15:16.700 --> 00:15:25.200
(this whole thing, AC) and the segment of the hypotenuse adjacent to that leg.
00:15:25.200 --> 00:15:36.100
"Adjacent" means "next to"; so then, to find the geometric mean, we need two numbers.
00:15:36.100 --> 00:15:44.500
The two numbers would be the hypotenuse and the segment of the hypotenuse adjacent to that leg--
00:15:44.500 --> 00:15:50.300
the part of the hypotenuse that is close to that leg that we are trying to find.
00:15:50.300 --> 00:16:01.300
Here, for example, we are looking for this leg; this is the geometric mean between...the two numbers would be
00:16:01.300 --> 00:16:10.700
the whole thing, the hypotenuse, AC, and the part of the hypotenuse (remember, this altitude
00:16:10.700 --> 00:16:23.500
can divide this hypotenuse into two parts, AD and BC) that is adjacent (meaning closest to, next) to that leg that we are trying to find.
00:16:23.500 --> 00:16:31.800
It would be right here--that part; that means that we make our proportion...
00:16:31.800 --> 00:16:40.500
Again, the geometric mean...BC goes here and here; or you can write BC, BC.
00:16:40.500 --> 00:16:52.600
And then, let's write that so that you know what we are looking for: BC is going to go here and here.
00:16:52.600 --> 00:17:07.700
And then, the two numbers that we are going to use are AC, the whole hypotenuse, and then this part right here; that is DC.
00:17:07.700 --> 00:17:09.800
OK, and then you would solve it that way.
00:17:09.800 --> 00:17:13.600
Now, let's say that we are looking for this leg.
00:17:13.600 --> 00:17:19.400
If this is the leg that we are looking for--let's say y--then it is going to be the geometric mean
00:17:19.400 --> 00:17:28.100
between, again, the whole thing, AC, and the part of the hypotenuse adjacent to that leg that we are looking for.
00:17:28.100 --> 00:17:36.000
In this case, if we are looking for y, this leg, we are not going to be using DC, because that is not the one adjacent to that.
00:17:36.000 --> 00:17:48.100
Then, we are going to be using AC and AD, because that is the part of the hypotenuse that is adjacent to this leg that we are looking for.
00:17:48.100 --> 00:18:02.900
Again, y, y...or you can write AB, AB...that is going to be the whole thing, AC, and AD.
00:18:02.900 --> 00:18:06.500
It is the geometric mean between those two numbers.
00:18:06.500 --> 00:18:17.700
Let's say that this whole thing is 20, and DC is 5.
00:18:17.700 --> 00:18:34.500
Now, let's look for BC; BC is the geometric mean between 20 and 5; we put 20 up here, x, and 5.
00:18:34.500 --> 00:18:43.700
It is going to be x² = 100; x = 10; so this right here is 10.
00:18:43.700 --> 00:19:00.600
To find y, if this is 5, and this whole thing is 20, I know that AD has to be 15, because 15 + 5 makes up this whole thing, which is 20.
00:19:00.600 --> 00:19:14.600
So, AC is 20, over y, equals y over this part, which is 15.
00:19:14.600 --> 00:19:41.500
y² (I am going to cross-multiply) equals...this would be 300, so y is going to be √300, which is 10√3.
00:19:41.500 --> 00:19:44.800
That is how you would find the geometric mean.
00:19:44.800 --> 00:19:56.700
Now, the one before (remember, the altitude one)--the altitude is the geometric mean between part and part of the hypotenuse.
00:19:56.700 --> 00:20:03.000
The leg is the geometric mean between the whole and this part close to it.
00:20:03.000 --> 00:20:09.700
Again, this leg is the geometric mean between the whole and the part close to it.
00:20:09.700 --> 00:20:15.000
Those are the two theorems, there; now let's go over our examples.
00:20:15.000 --> 00:20:20.500
Find the geometric mean between each pair of numbers.
00:20:20.500 --> 00:20:31.600
Find the geometric mean: to use our proportion, the geometric mean goes here and here;
00:20:31.600 --> 00:20:41.400
and then, the two numbers that we are going to use, that are given to us--those are the extremes, 8 and 11.
00:20:41.400 --> 00:20:47.300
Make sure that whatever the geometric mean is...it has to go here and here.
00:20:47.300 --> 00:21:26.700
Then, cross-multiply; I get x² = 88, so x = √88, and then that would simplify to...let's see...2√22.
00:21:26.700 --> 00:21:32.800
You can simplify it, if your teacher wants you to round it to several decimal places, or whatever it is.
00:21:32.800 --> 00:21:39.600
And you would have to use your calculator to figure that out in decimals.
00:21:39.600 --> 00:21:43.400
But this is how it would simplify.
00:21:43.400 --> 00:21:59.700
The next one: 5 and 2/3--again, 5/x = x/(2/3).
00:21:59.700 --> 00:22:05.400
Now, I am doing it this way, just so that it is easier to set up.
00:22:05.400 --> 00:22:13.300
But you also know that x is going to equal the square root of AB; so you can just use that if you want to.
00:22:13.300 --> 00:22:20.700
That is where you just take the two numbers, multiply it, and then take the square root of it to find the geometric mean.
00:22:20.700 --> 00:22:30.400
Or you can just set it up so that this is easier to understand, because you know that these two numbers make up the means.
00:22:30.400 --> 00:22:34.200
So, you know that whatever it says the mean is, you write that here.
00:22:34.200 --> 00:22:39.700
In this case, we are looking for it; that is why we have x's there.
00:22:39.700 --> 00:22:45.000
And then, the two numbers that are given would go there.
00:22:45.000 --> 00:23:00.500
Then, x² =...here, this becomes...5 times 2/3 is going to be 10/3, so x is the square root of 10/3.
00:23:00.500 --> 00:23:02.700
And then, just use your calculator for that.
00:23:02.700 --> 00:23:11.400
If you have to leave it in radical form, it would just be √10/√3.
00:23:11.400 --> 00:23:20.200
And then, I would have to rationalize this denominator, so it is going to be...see how √3/√3 is just equal to 1.
00:23:20.200 --> 00:23:28.400
So, this would be √30/3, because √3 times √3 equals 3.
00:23:28.400 --> 00:23:36.300
Now again, if you don't understand this, just go ahead and use your calculator; just do 10/3, and then you can just take the square root of that.
00:23:36.300 --> 00:23:48.400
Or if your calculator will allow you, just do the square root of 10/3.
00:23:48.400 --> 00:24:00.000
Name the three similar triangles: again, we have the altitude from the right angle of this big triangle.
00:24:00.000 --> 00:24:07.000
The altitude is from the right angle to the hypotenuse of the big triangle.
00:24:07.000 --> 00:24:10.800
We are going to name the three triangles that are similar.
00:24:10.800 --> 00:24:18.100
Now, again, we can start with whatever triangle we want.
00:24:18.100 --> 00:24:25.900
If you do this on your own, then your triangles will probably be different; it is probably going to be listed differently than how I list it.
00:24:25.900 --> 00:24:34.900
But that is fine, just as long as whatever you wrote, the three triangles, are corresponding with each other in the parts.
00:24:34.900 --> 00:24:38.200
The first triangle that I want to name is the big one.
00:24:38.200 --> 00:24:46.600
I want to name the big one first, and I am going to say that triangle...let's do the hypotenuse first:
00:24:46.600 --> 00:25:05.100
MOP...that P, my right angle, would be last; so MO, and then P is my right angle.
00:25:05.100 --> 00:25:24.300
And that is going to be similar to triangle...make sure that...MO is my hypotenuse, and then OP would be
00:25:24.300 --> 00:25:44.100
the short side; so triangle...and then, let's see, let's do this one...MPN--let's try that,
00:25:44.100 --> 00:25:51.300
because again, P is my right angle; so then, N, which is the right angle for this triangle, has to be listed last.
00:25:51.300 --> 00:25:55.300
So, is MP my hypotenuse?--yes, so that is right.
00:25:55.300 --> 00:26:10.900
How about PN--is PN my short side?--yes, so this is correct.
00:26:10.900 --> 00:26:21.100
And then, it is similar to...what is my other one?--the small triangle right here, right?
00:26:21.100 --> 00:26:30.100
So then, this is the right angle; that is going to go last; and the triangle's hypotenuse would be...
00:26:30.100 --> 00:26:44.500
let's see, OPN...OP is my hypotenuse; N is my right angle; and now, let's look at PN.
00:26:44.500 --> 00:26:55.200
For all of these other ones, OP was my short leg; PN was also my short leg of this triangle.
00:26:55.200 --> 00:27:08.700
Is PN the short side of this small triangle? It is actually not, so this is wrong.
00:27:08.700 --> 00:27:20.900
I know that OP is my hypotenuse, but then, instead of saying OPN, I would have to say PON.
00:27:20.900 --> 00:27:25.400
N is in the correct position, because that is the right angle, and that has to go last.
00:27:25.400 --> 00:27:33.800
I know that, instead of OP, like how we had it, it would have to go PO.
00:27:33.800 --> 00:27:47.500
PON: that way, PO is my hypotenuse, and then ON is my short side.
00:27:47.500 --> 00:27:50.200
Those are my three triangles that are similar.
00:27:50.200 --> 00:28:00.500
Now, again, if you had your first triangle listed out differently, and if you used another triangle, that is fine;
00:28:00.500 --> 00:28:07.800
just make sure that the other two triangles that you list out are similar to that triangle.
00:28:07.800 --> 00:28:13.800
It is always easiest to just maybe write some symbols like this, like how I did it.
00:28:13.800 --> 00:28:23.800
And then, just make sure that you know that your hypotenuse has to go hypotenuse, short leg, long leg, right angle...
00:28:23.800 --> 00:28:31.800
use those to help you list it in the corresponding order.
00:28:31.800 --> 00:28:34.600
The next example: we are going to find the values of x and y.
00:28:34.600 --> 00:28:40.300
We are actually going to do a few of these for the next example, too.
00:28:40.300 --> 00:29:03.600
For the first one, let's see: it is a right angle's altitude; that means that I know that this altitude is the geometric mean between these parts of the hypotenuse.
00:29:03.600 --> 00:29:17.400
So, that means that the altitude is the geometric mean, x and x, between 4 and 7.
00:29:17.400 --> 00:29:30.400
Solve this out; this is going to be x² = 28, so x is √28; does that simplify?
00:29:30.400 --> 00:29:38.800
Let's see: it is going to be 2√7.
00:29:38.800 --> 00:29:43.700
Again, if you want to change it to decimals, then use your calculator.
00:29:43.700 --> 00:29:47.400
2√7 is this value right here.
00:29:47.400 --> 00:29:56.600
And then, to find y, you can do two things: you can use that second theorem
00:29:56.600 --> 00:30:06.900
that says that the leg of the big triangle is the geometric mean between the whole hypotenuse
00:30:06.900 --> 00:30:18.800
and the part that is adjacent to it, and solve it out this way; or you can just, using this right triangle,
00:30:18.800 --> 00:30:25.200
now that you know what this side is, and you have this side as 7, use the Pythagorean theorem.
00:30:25.200 --> 00:30:35.500
This would be a² + b² = c² (or y²).
00:30:35.500 --> 00:30:39.700
Either way, it does not matter; you will still get the same answer.
00:30:39.700 --> 00:30:46.700
Let's just go ahead and solve it this way, with the geometric mean.
00:30:46.700 --> 00:30:56.600
So, we are going to write y and y there; this is the geometric mean between the whole thing--
00:30:56.600 --> 00:31:04.300
what is the whole thing?--4 + 7 is the whole thing; the whole hypotenuse would be 11;
00:31:04.300 --> 00:31:11.100
and then, the part of the hypotenuse that is close to this leg that we are looking for is 7.
00:31:11.100 --> 00:31:25.900
So, y² = 77; y = √77; and that is it; so we have here and here.
00:31:25.900 --> 00:31:36.600
The next one: Let's see, this altitude is the geometric mean between this part and this part.
00:31:36.600 --> 00:31:40.800
Now, this one is given to me; I am not looking for the altitude.
00:31:40.800 --> 00:31:44.200
But the theorem does not change; the theorem stays the same.
00:31:44.200 --> 00:31:50.000
It is still saying that the altitude is a geometric mean between this part and this part.
00:31:50.000 --> 00:31:57.600
So, when I write it in my proportion, I still have to keep this altitude, whatever it is,
00:31:57.600 --> 00:32:14.400
whether it is x or whether it is given--I still have to write it as my mean, there, and then between what?-- between this, which is x, and 5x.
00:32:14.400 --> 00:32:23.200
So, it still stays the same, just like this; whatever this is right here, that is going to go here and here.
00:32:23.200 --> 00:32:30.400
In the same way, whatever the altitude is, the altitude is the geometric mean; so then, that is going to go here and here.
00:32:30.400 --> 00:32:44.000
When you solve it out, it is going to be 25² = 5x²; you are just multiplying through.
00:32:44.000 --> 00:32:54.700
And then, 25² is 625; that is equal to 5x².
00:32:54.700 --> 00:33:29.000
If you divide the 5, that is going to give you 125 = x², so √125 is going to be equal to x, which is 5√5.
00:33:29.000 --> 00:33:40.100
And that is a 5 right there; so 5√5 is x, so this is going to be 5√5.
00:33:40.100 --> 00:33:52.300
This is going to be 5 times x, which is 5 times 5√5, is 25√5.
00:33:52.300 --> 00:34:02.100
And then, y: again, you can use the Pythagorean theorem: 25² + (5√5)² = y².
00:34:02.100 --> 00:34:17.900
Or you can just use this: y is a geometric mean between the whole thing--
00:34:17.900 --> 00:34:28.600
the whole thing would be 25√5 + 5√5; that is going to be 30√5--
00:34:28.600 --> 00:34:36.400
and this part that is close to the leg, 5√5.
00:34:36.400 --> 00:35:03.900
y² is equal to...this is going to be 30 times 5, is going to be 150; and then, √5 times √5 becomes 5.
00:35:03.900 --> 00:35:13.800
So then, it is 150 times 5; so again, √5 times √5 is (√5)².
00:35:13.800 --> 00:35:19.100
Remember: that square root and the square cancel each other out, so it just becomes 5.
00:35:19.100 --> 00:35:58.000
And then, this becomes 750; and then, y becomes √750, and to simplify that out, it becomes...
00:35:58.000 --> 00:36:31.400
let's see...25...OK, well, here, y equals 5√30; I believe that is correct.
00:36:31.400 --> 00:36:36.100
Or you can just, again, change it to decimals with your calculator.
00:36:36.100 --> 00:36:45.300
It is kind of a large number, so just go ahead and use your calculator.
00:36:45.300 --> 00:36:53.900
And that is it for this problem; here is x, and here is y, for this.
00:36:53.900 --> 00:36:56.700
Now, we are going to go on to the next problem; and for the next one,
00:36:56.700 --> 00:37:03.400
we are going to go over a couple more, just so you can get more familiar with these types of problems.
00:37:03.400 --> 00:37:16.600
Again, here we have the geometric mean, the altitude being the geometric mean between this part and this part.
00:37:16.600 --> 00:37:30.500
So then, the geometric mean is going to go here and here, between these two numbers, x and 8.
00:37:30.500 --> 00:37:48.000
8x = 9; x is equal to 9/8; and then, for the y, again, you can use the same concept,
00:37:48.000 --> 00:37:54.200
the theorem that says that this is the geometric mean between the whole thing and then this part.
00:37:54.200 --> 00:37:59.700
Or we can use the Pythagorean theorem; let's go ahead and just use the Pythagorean theorem this time.
00:37:59.700 --> 00:38:08.400
So then, a², the leg squared, plus the other leg squared, is equal to the hypotenuse squared.
00:38:08.400 --> 00:38:13.900
3² + 8² = y².
00:38:13.900 --> 00:38:31.600
3² is 9, plus 64, is equal to y²; 73 = y²; y = √73.
00:38:31.600 --> 00:38:39.700
And you can just leave it like that; that does not simplify.
00:38:39.700 --> 00:38:51.400
The next one: the same thing: this is the geometric mean between this part and that part.
00:38:51.400 --> 00:39:18.800
So then, 12...12...the geometric between 8 and x...8x = 144; divide the 8; x is equal to 18.
00:39:18.800 --> 00:39:38.300
And then, here, to find the y, the same thing happens.
00:39:38.300 --> 00:39:45.600
We are going to say that this...we know that, since this is 18, we can use the Pythagorean theorem.
00:39:45.600 --> 00:39:49.700
12² + 18² is equal to y².
00:39:49.700 --> 00:40:00.600
Or we can use the theorem; let's just use the theorem: y...y; that is the geometric mean between the whole thing...
00:40:00.600 --> 00:40:17.800
18 + 8 is 26; that is the whole thing; this whole thing would be 26...and then this part that is close to it is 18.
00:40:17.800 --> 00:40:27.200
Those are my extremes, and then my mains would be this: y² is equal to--
00:40:27.200 --> 00:40:39.800
and you can just use your calculator for this part...it is 26 times 18, which is 468;
00:40:39.800 --> 00:40:56.100
and then, y is equal to the square root of 468, and again, you can just use your calculator to simplify that out.
00:40:56.100 --> 00:40:59.000
That is it for this lesson; thank you for watching Educator.com.