Geometry Geometric Mean

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We are going to go over the geometric mean for our next lesson.0002

For the geometric mean, here, we have a/x, which is equal to x/b.0006

Now, if you remember, a few lessons ago, we went over ratios and proportions.0019

When we have a proportion, we have two ratios that are equal to each other.0025

Here, these numbers right here are called the means, and these numbers here are called extremes.0032

So, when we talk about the geometric mean, we are talking about this number and this number.0049

The geometric mean between two positive numbers, a and b, is x.0060

If we are given two numbers, a and b, those numbers would be considered the extremes.0067

And then, x, which is the geometric mean, is going to go here and here.0073

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.0082

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...0094

Now, actually, before we continue with that problem, let's go back to this.0114

If we cross-multiply, then we are going to get x² = a times b; then how would you find x?0121

x is going to be the square root of ab.0134

The geometric mean x is going to be the square root of this number times that number.0140

This is how you would find the geometric mean between the two numbers.0148

Now, back to this: 3 and 8--find the geometric mean between 3 and 8.0152

What you can do is go ahead and solve it like this: plug it into here and find the square root of 3 times 8.0159

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 given0168

different versions of this problem, like if you are given the geometric mean and asked to find one of these numbers.0179

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.0186

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.0196

And then, 3 and 8 are going to go here and here; so this is how you would set it up.0209

Find the geometric mean--that means that these numbers right here are what you are looking for.0216

To solve it out, cross-multiply: x² is equal to 3 times 8, which is 24, so x is equal to √24.0220

And that is going to be 2√6, or you can use your calculator to change that to a decimal.0235

But this would be the answer, right here: this is the geometric mean between 3 and 8.0247

Again, set it up as a proportion; put the two numbers up here and down there.0254

Make sure that the geometric mean goes here and here, because those are the means; and then, just cross-multiply to solve.0262

Similar triangles: now, here we have a right triangle, and remember: an altitude is a segment0274

starting from a vertex and going to the side opposite that vertex so that it is perpendicular to it.0286

If we draw an altitude from this vertex (and it has to be drawn from that right angle)--0299

so then here is our right angle, and you are going to draw an altitude so that it is perpendicular to the side opposite--0307

in this case, it is the hypotenuse, because it is coming from that right angle--then the two triangles formed0318

are similar to the given triangle and to each other.0325

Before this altitude was drawn, we only had one triangle.0330

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.0335

Now, this theorem is saying that, once that altitude is drawn from that right angle to the hypotenuse, all three triangles are now similar.0345

Remember: similar triangles are triangles that have congruent angles, but proportional sides.0356

"Similar" just means that they have the same shape, but a different size; that is "similar" or "similarity."0368

So, here we know that we have three triangles--the same shape, but slightly different sizes.0377

To state all three triangles so that the corresponding parts are in order, so that we can say that they are similar,0386

I can name all three; it doesn't matter which order, so if I want to say the big one first:0397

the big one is going to be triangle ABC; it is similar to...and then we are going to just name another triangle.0406

Now, remember: it has to be corresponding to the order; so AB in the big triangle0419

(and that is kind of hard to see)...the easiest way to see their corresponding parts is to look at it from long leg0429

to short leg, and then hypotenuse; those are the three parts that make up a right triangle.0445

It is always easiest to determine whatever you are stating--which side it would be considered from that triangle.0453

And then, you can just look at the corresponding part of the other triangle.0464

So, AB from the big triangle is considered the long leg; this would be AB.0468

AB is the long leg; BC is the short leg; and then, AC is the hypotenuse.0479

If AB is the long leg, then I have to mention that one first.0487

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.0491

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.0499

I am going to label this D; so remember, I have to state the long first.0510

The long leg from this triangle would be AD, because here is the right angle; AD is the long leg.0516

Now, does it matter if I say AD or DA?0525

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.0528

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.0540

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.0556

For my second triangle that I am naming, AD is my long leg; BD is my short leg; and AB is my hypotenuse.0571

And then, see how D is the right angle; the middle angle that I am listing is the right angle.0583

So then, the third triangle has to be in the same order, so from this one right here,0593

which one is my long leg? Which one is my short leg? And which one is my hypotenuse?0600

My long leg from here...this is the right angle...would be BD or DB.0606

Now, from this triangle, where is my right angle?--this one, D.0613

I know that that letter D is going to go in the middle.0618

So then, BD is my long leg; D is my right angle; C is my missing vertex.0622

BD is my long leg; DC is my short leg; and BC is my hypotenuse.0634

So now, I have listed all three triangles within their corresponding parts.0642

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.0653

Here, we have a right triangle, again, with the altitude drawn from the right angle of that triangle down to the hypotenuse.0672

When you have this diagram, the measure of the altitude, BD, is going to be the geometric mean between AD and DC.0685

Remember how, when we went over geometric mean, we found the geometric mean between two numbers.0710

So then, the two numbers that are given are a and b, and then, this is a geometric mean and a geometric mean.0719

This right here, BD, would be the geometric mean between the two parts of the hypotenuse.0731

Just to read this to you: the measures of the altitude drawn from the vertex of the right angle of the right triangle0739

to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.0748

So, because this right here--see how it splits up the hypotenuse into two parts, AD and DC--those two parts,0756

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.0765

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.0778

Then, AD would go up here, and DC would go right there; that is like a and b--those are the two numbers.0788

And then, the geometric mean, which is the altitude, BD, is going to go here and here.0798

So, whenever they say that something is the geometric mean, that something goes in here and here,0808

because remember, again: these two would be the means.0816

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 5.0823

So then, we are trying to find the altitude.0838

Now, we know, because of this theorem, that the altitude, BD, is the geometric mean between AD and DC, these two parts.0840

And this would be the geometric mean; so to solve it out, I am going to make it into my proportion.0855

The geometric mean is BD; that is going to go here and here.0863

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 10.0867

So then, this right here...the measure of the altitude is 10.0879

Now, one more: here, for this one, if the altitude is drawn to the hypotenuse of a right triangle,0888

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)0897

is the geometric mean between the measures of the hypotenuse0909

(this whole thing, AC) and the segment of the hypotenuse adjacent to that leg.0917

"Adjacent" means "next to"; so then, to find the geometric mean, we need two numbers.0925

The two numbers would be the hypotenuse and the segment of the hypotenuse adjacent to that leg--0936

the part of the hypotenuse that is close to that leg that we are trying to find.0944

Here, for example, we are looking for this leg; this is the geometric mean between...the two numbers would be0950

the whole thing, the hypotenuse, AC, and the part of the hypotenuse (remember, this altitude0961

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.0971

It would be right here--that part; that means that we make our proportion...0983

Again, the geometric mean...BC goes here and here; or you can write BC, BC.0992

And then, let's write that so that you know what we are looking for: BC is going to go here and here.1000

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.1012

OK, and then you would solve it that way.1028

Now, let's say that we are looking for this leg.1030

If this is the leg that we are looking for--let's say y--then it is going to be the geometric mean1033

between, again, the whole thing, AC, and the part of the hypotenuse adjacent to that leg that we are looking for.1039

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.1048

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.1056

Again, y, y...or you can write AB, AB...that is going to be the whole thing, AC, and AD.1068

It is the geometric mean between those two numbers.1083

Let's say that this whole thing is 20, and DC is 5.1086

Now, let's look for BC; BC is the geometric mean between 20 and 5; we put 20 up here, x, and 5.1098

It is going to be x² = 100; x = 10; so this right here is 10.1114

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 20.1124

So, AC is 20, over y, equals y over this part, which is 15.1140

y² (I am going to cross-multiply) equals...this would be 300, so y is going to be √300, which is 10√3.1154

That is how you would find the geometric mean.1181

Now, the one before (remember, the altitude one)--the altitude is the geometric mean between part and part of the hypotenuse.1185

The leg is the geometric mean between the whole and this part close to it.1197

Again, this leg is the geometric mean between the whole and the part close to it.1203

Those are the two theorems, there; now let's go over our examples.1210

Find the geometric mean between each pair of numbers.1215

Find the geometric mean: to use our proportion, the geometric mean goes here and here;1220

and then, the two numbers that we are going to use, that are given to us--those are the extremes, 8 and 11.1231

Make sure that whatever the geometric mean is...it has to go here and here.1241

Then, cross-multiply; I get x² = 88, so x = √88, and then that would simplify to...let's see...2√22.1247

You can simplify it, if your teacher wants you to round it to several decimal places, or whatever it is.1287

And you would have to use your calculator to figure that out in decimals.1293

But this is how it would simplify.1299

The next one: 5 and 2/3--again, 5/x = x/(2/3).1303

Now, I am doing it this way, just so that it is easier to set up.1320

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.1325

That is where you just take the two numbers, multiply it, and then take the square root of it to find the geometric mean.1333

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.1341

So, you know that whatever it says the mean is, you write that here.1350

In this case, we are looking for it; that is why we have x's there.1354

And then, the two numbers that are given would go there.1360

Then, x² =...here, this becomes...5 times 2/3 is going to be 10/3, so x is the square root of 10/3.1365

And then, just use your calculator for that.1380

If you have to leave it in radical form, it would just be √10/√3.1383

And then, I would have to rationalize this denominator, so it is going to be...see how √3/√3 is just equal to 1.1391

So, this would be √30/3, because √3 times √3 equals 3.1400

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.1408

Or if your calculator will allow you, just do the square root of 10/3.1416

Name the three similar triangles: again, we have the altitude from the right angle of this big triangle.1428

The altitude is from the right angle to the hypotenuse of the big triangle.1440

We are going to name the three triangles that are similar.1447

Now, again, we can start with whatever triangle we want.1451

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.1458

But that is fine, just as long as whatever you wrote, the three triangles, are corresponding with each other in the parts.1466

The first triangle that I want to name is the big one.1475

I want to name the big one first, and I am going to say that triangle...let's do the hypotenuse first:1478

MOP...that P, my right angle, would be last; so MO, and then P is my right angle.1486

And that is going to be similar to triangle...make sure that...MO is my hypotenuse, and then OP would be1505

the short side; so triangle...and then, let's see, let's do this one...MPN--let's try that,1524

because again, P is my right angle; so then, N, which is the right angle for this triangle, has to be listed last.1544

So, is MP my hypotenuse?--yes, so that is right.1551

How about PN--is PN my short side?--yes, so this is correct.1555

And then, it is similar to...what is my other one?--the small triangle right here, right?1571

So then, this is the right angle; that is going to go last; and the triangle's hypotenuse would be...1581

let's see, OPN...OP is my hypotenuse; N is my right angle; and now, let's look at PN.1590

For all of these other ones, OP was my short leg; PN was also my short leg of this triangle.1604

Is PN the short side of this small triangle? It is actually not, so this is wrong.1615

I know that OP is my hypotenuse, but then, instead of saying OPN, I would have to say PON.1629

N is in the correct position, because that is the right angle, and that has to go last.1641

I know that, instead of OP, like how we had it, it would have to go PO.1645

PON: that way, PO is my hypotenuse, and then ON is my short side.1654

Those are my three triangles that are similar.1667

Now, again, if you had your first triangle listed out differently, and if you used another triangle, that is fine;1670

just make sure that the other two triangles that you list out are similar to that triangle.1680

It is always easiest to just maybe write some symbols like this, like how I did it.1688

And then, just make sure that you know that your hypotenuse has to go hypotenuse, short leg, long leg, right angle...1694

use those to help you list it in the corresponding order.1704

The next example: we are going to find the values of x and y.1712

We are actually going to do a few of these for the next example, too.1714

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.1720

So, that means that the altitude is the geometric mean, x and x, between 4 and 7.1743

Solve this out; this is going to be x² = 28, so x is √28; does that simplify?1757

Let's see: it is going to be 2√7.1770

Again, if you want to change it to decimals, then use your calculator.1779

2√7 is this value right here.1784

And then, to find y, you can do two things: you can use that second theorem1787

that says that the leg of the big triangle is the geometric mean between the whole hypotenuse1796

and the part that is adjacent to it, and solve it out this way; or you can just, using this right triangle,1807

now that you know what this side is, and you have this side as 7, use the Pythagorean theorem.1819

This would be a² + b² = c² (or y²).1825

Either way, it does not matter; you will still get the same answer.1835

Let's just go ahead and solve it this way, with the geometric mean.1840

So, we are going to write y and y there; this is the geometric mean between the whole thing--1847

what is the whole thing?--4 + 7 is the whole thing; the whole hypotenuse would be 11;1856

and then, the part of the hypotenuse that is close to this leg that we are looking for is 7.1864

So, y² = 77; y = √77; and that is it; so we have here and here.1871

The next one: Let's see, this altitude is the geometric mean between this part and this part.1886

Now, this one is given to me; I am not looking for the altitude.1896

But the theorem does not change; the theorem stays the same.1901

It is still saying that the altitude is a geometric mean between this part and this part.1904

So, when I write it in my proportion, I still have to keep this altitude, whatever it is,1910

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.1917

So, it still stays the same, just like this; whatever this is right here, that is going to go here and here.1934

In the same way, whatever the altitude is, the altitude is the geometric mean; so then, that is going to go here and here.1943

When you solve it out, it is going to be 25² = 5x²; you are just multiplying through.1950

And then, 25² is 625; that is equal to 5x².1964

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√5.1975

And that is a 5 right there; so 5√5 is x, so this is going to be 5√5.2009

This is going to be 5 times x, which is 5 times 5√5, is 25√5.2020

And then, y: again, you can use the Pythagorean theorem: 25² + (5√5)² = y².2032

Or you can just use this: y is a geometric mean between the whole thing--2042

the whole thing would be 25√5 + 5√5; that is going to be 30√5--2058

and this part that is close to the leg, 5√5.2068

y² is equal to...this is going to be 30 times 5, is going to be 150; and then, √5 times √5 becomes 5.2076

So then, it is 150 times 5; so again, √5 times √5 is (√5)².2104

Remember: that square root and the square cancel each other out, so it just becomes 5.2114

And then, this becomes 750; and then, y becomes √750, and to simplify that out, it becomes...2119

let's see...25...OK, well, here, y equals 5√30; I believe that is correct.2158

Or you can just, again, change it to decimals with your calculator.2191

It is kind of a large number, so just go ahead and use your calculator.2196

And that is it for this problem; here is x, and here is y, for this.2205

Now, we are going to go on to the next problem; and for the next one,2214

we are going to go over a couple more, just so you can get more familiar with these types of problems.2217

Again, here we have the geometric mean, the altitude being the geometric mean between this part and this part.2223

So then, the geometric mean is going to go here and here, between these two numbers, x and 8.2236

8x = 9; x is equal to 9/8; and then, for the y, again, you can use the same concept,2250

the theorem that says that this is the geometric mean between the whole thing and then this part.2268

Or we can use the Pythagorean theorem; let's go ahead and just use the Pythagorean theorem this time.2274

So then, a², the leg squared, plus the other leg squared, is equal to the hypotenuse squared.2280

3² + 8² = y².2288

3² is 9, plus 64, is equal to y²; 73 = y²; y = √73.2294

And you can just leave it like that; that does not simplify.2311

The next one: the same thing: this is the geometric mean between this part and that part.2320

So then, 12...12...the geometric between 8 and x...8x = 144; divide the 8; x is equal to 18.2331

And then, here, to find the y, the same thing happens.2359

We are going to say that this...we know that, since this is 18, we can use the Pythagorean theorem.2378

12² + 18² is equal to y².2385

Or we can use the theorem; let's just use the theorem: y...y; that is the geometric mean between the whole thing...2390

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 18.2400

Those are my extremes, and then my mains would be this: y² is equal to--2418

and you can just use your calculator for this part...it is 26 times 18, which is 468;2427

and then, y is equal to the square root of 468, and again, you can just use your calculator to simplify that out.2440

That is it for this lesson; thank you for watching Educator.com.2456