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### Triangle Inequality

- Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side

### Triangle Inequality

A side of one triangle can be smaller than the difference of other two sides of the triangle.

8,15,9

6, 4, 2

- 13 + 18 = 31
- 18 − 13 = 5

- 3 + 5 = 8
- 5 − 3 = 2

―AC is _____ larger than 4.

If ―AB = 9, ―AC = 12, then ―BC can be 15.

A(4, 5), B(1, 3), C(5, 8).

- ―AB = √{(1 − 4)
^{2}+ (3 − 5)^{2}} = √{9 + 4} = √{13} ≈ 3.6 - ―AC = √{(5 − 4)
^{2}+ (8 − 5)^{2}} = √{1 + 9} = √{10} ≈ 3.2 - ―BC = √{(5 − 1)
^{2}+ (8 − 3)^{2}} = √{16 + 25} = √{41} ≈ 6.4

In a triangle, if the measurement of a side is 18, then the measurements of the other two sides can be 5 and 25.

In a triangle, if the measurement of a side is 25, then the measurements of other two sides can be 21 and 30.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Triangle Inequality

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Triangle Inequality Theorem 0:05
- Triangle Inequality Theorem
- Triangle Inequality Theorem 4:22
- Example 1: Triangle Inequality Theorem
- Example 2: Triangle Inequality Theorem
- Extra Example 1: Determine if the Three Numbers can Represent the Sides of a Triangle 12:00
- Extra Example 2: Finding the Third Side of a Triangle 13:34
- Extra Example 3: Always True, Sometimes True, or Never True 18:18
- Extra Example 4: Triangle and Vertices 22:36

### Geometry Online Course

### Transcription: Triangle Inequality

*Welcome back to Educator.com.*0000

*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 B*0401

*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 (x _{2} - x_{1}), or you could do (x_{1} - x_{2})--*1400

*it doesn't matter--squared, plus (y _{2} - y_{1}), 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 (x _{1},y_{1}) and (x_{2},y_{2}).*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 (x _{1}), 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

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