Geometry Triangle Inequality

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In the next lesson, we are going to go over the triangle inequality theorem.0002

Now, remember: inequality is when we are dealing with greater than/less than/greater than or equal to/less than or equal to.0006

When we are talking about triangle inequality, we are talking about these; we are comparing different sides and angles together.0019

So, the triangle inequality theorem says that the sum of the lengths of any two sides of a triangle is greater than the length of the third side,0028

meaning (now, we have three sides in a triangle) the lengths of any two sides has to be greater than the third side.0039

So, if I add up side AB with side AC, then this has to be greater than BC.0050

And it just means, for example, that...let's say that point A is your home; point A is where you live--that is your home.0063

Let's say you walk to school, and point B is school.0079

Every day, you walk from point A to point B: you walk from home to school in the morning.0088

And one day, you decide, "Oh, I want to stop by the market to get breakfast," or something.0095

You want to stop by somewhere; let's say you want to stop by your friend's house to pick up your friend, so that you can walk together.0102

This, let's say, is the market, or friend's house, or wherever.0108

From home, you decide to go to the market, and then go to the school.0116

The distance here, from home to school, is, let's say, x; now, if the distance from home to the market is y,0124

and the distance from the market to school is z; if you are going to walk from home to the market, and then to school,0139

you can agree that that is going to be a further distance than if you just walked from home directly to school.0149

From home to the market to school is going to be greater than just from home to school; and that is what this inequality is saying.0157

It is saying that if you are going to add up any two sides, that is going to be greater than just directly from a point to another point,0166

because it is like you are detouring--think of a detour: whenever you detour, it is a further distance.0175

So, again, from home to school is going to be a shorter distance, because it is direct; you are going straight to that point.0182

It is going to be shorter; this alone is going to be less than home to market to school.0191

Or you can say that home to the market, or home (point A) to point C, then to point B, is going to be greater than just going from point A to point B.0200

That is the triangle inequality theorem: the length of two sides, if you add it up, is going to be greater than the third side by itself.0210

Back to this right here: AB + AC is greater than BC, or I can say AB + BC is greater than AC.0223

And the other two sides, AC + BC, are going to be greater than AB.0242

You stop by a point; two of the lengths are going to be greater than the third length by itself.0252

And that is the triangle inequality theorem.0259

Here is a triangle, right here; and let's say that AC is 10, and BC is 8; I want to find possible lengths for AB.0265

Now, I know that the sum of any two sides has to be greater than the third side.0281

So, I know that this cannot be 1; why?--because if this is 1, well, then, 1 + 8 is 9, and then this is 10;0285

so then, these added together have to be greater than the 10.0299

So then, this can't be 1, because AB + BC will equal 9, and that is not greater than AC.0303

It can't be 2, because if it is 2, then 2 + 8 is 10, and that is the same as AC.0311

It can't be the same--you can't say that if you are going to walk from point A to point C,0322

that is going to be exactly the same distance as if you walked from point A to B to C.0326

So, you can't say that this is 2; now, can I say that it is 3?0331

Yes, I can say that it is 3.0336

Now, I can say that it is 3, because if this is 3, and this is 8, then that is 11, and that is greater than this.0340

So, even though I can't say that it is 2, I can say that it has to be greater than 2.0351

That means that it can't be 2, but it could be greater than 2.0360

So, it could be 2.5, because 2.5 + 8 is 10.5, which is still greater than AC.0364

So, it works as long as it is greater than 2...but there is a maximum number, too,0372

because you are not just adding up these two sides to have it be greater than 10.0378

It is just like the previous slide--remember: we went over AB + BC is greater than AC, and BC + AC is greater than AB, and then the other one.0386

This can't be 20, because if I add up AC + BC, that is 18; and if this is 20, then you are saying that point A to point B0401

is farther than if you go from point A to point C to point B, if you are saying that this is 20.0415

Let's say that this is 20 miles; if this is 20 miles, then A to C to B can't be 18.0423

So, we know that it has to be greater than 2; but then, if I add up AC and BC, then that becomes 18.0432

So then, I know that this has to be shorter than 18, because if this your home to school,0444

then it obviously has to be less than 18 miles; you are not going to be walking 18 miles to school...but anywhere;0457

from point A to point B has to be a shorter distance than if you are going to go from point A to somewhere else, and then back to point B.0466

So then, if this is 18, then this has to be shorter than 18; so then, it has to be less than 18.0476

BC has to be greater than 2, but less than 18; those are the possible lengths of AB.0488

Oh, wait; I wrote BC, and I need to write AB; so again, AB has to be greater than 2, but it has to be less than 18.0503

So, it has to be any number between these two; and this is how you would write it.0513

When you write an inequality, if you want to say that it is going to be between two numbers, you can say AB,0519

or x, or whatever it is, has to be greater than 2, and less than 18; and that is how you state AB being between 2 numbers.0527

How did we come up with these numbers, again?0543

I know that AB + BC has to be greater than 10; so then, this has to be greater than 2, because 2 and 8 make 10.0545

So then, it has to be greater than 2 so that this will be greater than 10.0553

The same thing here: BC and AC together have to be greater than AB, so if they make 18, then this has to be less than 18.0558

Those are two possible numbers that it can be between.0578

Now, let's say that I give you three numbers; let's say this is 15; this is 7; and this is 6.0580

And I ask you, "Can these three be the lengths of these sides?"0603

Your answer is going to be yes or no, or true and false.0609

Again, we have to see that the sum of any two sides has to be greater than the third side.0614

What I can do here, to see if it can be true, is: I am going to add up the two smallest sides,0621

because obviously, if this is the greatest side, if I add this to any one of these, it is going to be greater than the third.0626

The biggest plus anything is going to be bigger than the smallest, and the biggest plus the other small one is going to be greater.0634

We don't have to actually add up all pairs; instead, just add up the smallest lengths--that is 6 and 7.0641

6 and 7 (AB + BC) should be greater than AC; AB is 6, plus BC is 7; is that greater than (this is a question mark) 15?0654

13 (6 + 7 is 13)...is it greater than 15? No, it is not.0676

So, this one is false, because if you add up 6 and 7, it makes 13; that is smaller than 15, so this one is "no."0683

And it is just common sense; if you were to walk from point A to point C, then for you to stop by point B,0695

before going to point C--it can't be that this is going to be shorter, or else you are going to always walk the long way!0704

Again, this is false; this is not true.0714

Let's go over some of our examples: the first one: Determine if the three numbers can represent the sides of a triangle.0721

We have 6, 10, and 4; and we are going to see if these can be the lengths of the three sides of our triangle.0728

And again, if I take the biggest one, and I add it up to one of the two numbers, we are obviously going to have a bigger number.0739

So, we don't have to deal with the biggest number; instead, take the two smallest numbers, which are 6 and 4;0748

add them up, and just see if they are going to be greater than the third one.0754

So, 6 + 4 is 10; you know that 6 + 4 should be greater than 10, but is it?0758

No, 10 is not greater than 10; so this one is "no"; it cannot be the three sides of a triangle.0770

And the second one: 18 + 22 (those are the two smaller sides) should be greater than 45; 18 + 22 is 40.0780

We know that 40 is not greater than 45, so then this one is also "no."0801

These cannot be the sides of the triangle, the lengths of the triangle.0808

If two of the measures of sides are given, between what two numbers must the measure of the third side fall?0816

Again, we are given two sides, and we want to know possible measures for the third side.0823

And we are going to write that as an inequality.0832

18 and 22: if I have a triangle, and one side is 18, and the other side is 22, the let's call that third side x.0836

What I can do is: I know that 18 + x has to be greater than 22.0847

So then, x has to be (if I subtract 18) greater than 4.0861

I also know that 22 + x has to be...0868

or let's say 18 + 22 has to be greater than x; so this is going to be 40, so x has to be smaller than 40,0883

because again, these two have to be greater than 22, and those two have to be greater than x.0902

That means that x has to be greater than 4, but then x has to be smaller than 40.0912

Now, sometimes this could be hard to read, because x is on the other side.0917

Think of it as 40 > x, so then, let's do it by height; if Sarah is taller than Tim,0923

then I have to say that Tim is shorter than Sarah.0939

So, if 40 is greater than x, that means that x is less than, or smaller or shorter than, 40.0945

So, if it helps, you can flip this around; it is less than 40.0954

So then again, if you want to write between two numbers, you are going to write that x is going to be greater than 4.0964

Remember: if 4 is smaller than x, then x has to be bigger than 4.0972

And then, x is also less than 40--between 40 and 4.0976

And just to show you a shortcut to find this: all you do is subtract the two numbers to get the smallest number possible...0984

or not even the smallest; it just has to be greater than that number.0996

And then, you add the two numbers, and it has to be smaller than that number.0999

So then, x has to be bigger than when you subtract the two numbers, and it has to be smaller than when you add the two numbers.1005

Here, x has to be greater than...we will subtract them and get 88.1015

And then, you add them; it has to be 92, so it has to be smaller than 92.1024

The same thing with D and E: x has to be greater than D - E, and then...1035

Now, just in case E is, let's say, bigger than D, I am going to put this in absolute value.1054

And then, it is less than D + E; and with this one, I don't have to put an absolute value, because there is no chance that it could be negative.1070

If E is bigger than D, then if you subtract them, they are going to become negative--so it is just the absolute value, whatever their difference is.1081

x has to be greater than that number, and then it has to be smaller than D + E.1092

The next example: Indicate whether each statement is always true, sometimes true, or never true.1100

Here we have, let's see, a quadrilateral; we have several different triangles: 1, 2, 3; and then, we have another one right here--4.1106

We are going to see whether each one could be always true, sometimes true, or never true.1121

BE: if this is 10, and this is 4, is BE going to be greater than 14?1128

Well, if we look at this triangle right here, any two sides have to be greater than the third side.1137

This is 10; this is 4; that is 14; that means that this has to be less than 14.1143

But it says that it is greater than 14; so then, this one is never true, because,1150

if this is, let's say, 20--it is saying BE has to be greater than 14; 20 is greater than 14--1160

then you are saying that from point E to B is 20, and if you go to E to A, then to B, then it is 14.1169

Isn't it true that, if you are going to go from this point directly to this point, that should be the fastest way,1182

instead of having to cut, detour, and then go that way?1187

This one is never true; this cannot be greater than 14.1192

BC (the next one) is 13; BD is 5; well, it doesn't look like it could be 5, because it looks like it is going to be longer than 5.1199

BD looks like it is going to be longer than BC.1214

But do not assume that it is automatically going to be that way, just because the diagram looks like it.1218

We are just going to base it on facts.1227

Now, BC is 13; CD is 9; can BD be 5?1229

Well, again, here is a triangle; you are going to add up the two shortest sides, so 5 + 9 is 14; is that greater than the third side, 13?1237

Yes; that doesn't mean that it is going to be 5, but it could be 5.1248

It could be, because by the triangle inequality theorem, just as long as two sides add up to be greater than the third side, it will work.1253

But it doesn't mean that it is going to be 5.1263

This one is sometimes true, because it can work, but that doesn't mean that it is always going to be 5.1264

There are a lot of numbers that work; this is just one of them, so this is sometimes true.1270

And the third one: If angle A is a right angle (let me erase these, because those numbers don't apply to the next one),1277

then BD (this one right here) is going to be less than BA.1288

OK, just to draw it out to maybe make it a little easier to see: angle A is a right angle--that is angle A;1294

this is 10; this is 12, because it is the whole thing, if this is BD; BD is going to be less than BA.1305

No; remember: last lesson, we went over the theorem that says that the side opposite the greatest angle is going to be the longest side.1321

So, if this is a 90-degree angle, the hypotenuse is going to be the longest side of a right triangle.1337

So obviously, BD cannot be less than AB or BA; so this is never true.1342

The fourth example: We are going to determine whether it is possible to have a triangle with the given vertices.1358

Is it possible for my triangle to be on these coordinates?1368

And the way you figure that out is to find the distance between them and see if any of the two sides (the distance)...1376

if the sum of the two shortest sides is greater than the longest side (then it will work).1393

The distance formula, if you don't know, is (x₂ - x₁), or you could do (x₁ - x₂)--1400

it doesn't matter--squared, plus (y₂ - y₁), squared; and again, don't think that this is x squared;1409

and this is not y squared; it is saying the second x and the second y, the first x and the first y,1417

because you are finding the distance between two points; so then, you are going to have (x₁,y₁) and (x₂,y₂).1425

But it is not x squared; don't make that mistake--you are not squaring that number.1433

Let's say the distance between A and B first: for AB, you are going to have 1 (x₁), minus -2, squared,1440

plus 1 - 5, squared, equals...here this becomes a plus; that is 3 squared; that is 9, plus...that is -4 squared; that is 16,1453

which equals the square root of 25, which is 5.1472

Then, AC is going to be 1 - -5, squared, plus 1 - -4, squared.1477

So then again, this is plus; so I have 6 squared (that is 36), plus...this is also plus...so that is 5 squared; that is 25.1496

That is the square root of 61.1512

And then, BC is the last one: for BC, it is -2 - -5, squared, plus 5 - -4...1518

This becomes 3 squared (is 9), plus 9 squared (is 81); so here we have the square root of 90.1546

OK, now, if you have a calculator, you can use your calculator to change this to a decimal.1562

That way, you can compare these numbers with each other.1569

If not, I can just kind of estimate what decimal this is going to be.1572

If I know my perfect squares, I want to see what numbers this number falls between.1578

Well, let's see: 7 times 7 is 49, so the square root of 49 is 7; and then, the square root of 64 is 8.1584

It falls between those, but it falls...maybe a little bit between...right here, a little bit closer to the 64.1604

So, this is probably going to be around 7.7 or 8, or something like that; this is around 7.8, let's say.1613

This one is the same thing: 9 times 9 is 81, and then 10 times 10 is 100; it falls kind of right in the middle.1625

If this is 9 and this is 10, something that falls in the middle is going to be around maybe 9.5.1641

Notice how it is not an equals sign; it is squiggly--it is saying "approximately," "around 9.5."1650

The two shortest sides would be 5 and 7.8; now, I know that, if I add these two together, for sure I am going to get greater than 9.5.1657

So, 5 + 7.8 is about 12.8, and 12.8 is greater than 9.5; so this is "yes"; these vertices work; it is possible.1669

That is it for this lesson; thank you for watching Educator.com.1688