Video: The Triangle Inequality

Video Transcript

In this video, we’re going to look at the triangle inequality which tells us about an important relationship that must exist between the lengths of the sides in a triangle. So this is what the triangle inequality says. It says: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

So let’s have a look at this triangle that I’ve drawn here. What it means is if I take two sides, so let’s take 𝐴𝐵 and 𝐵𝐶, the sum, so I’m gonna add them together, so the sum of those two sides 𝐴𝐵 and 𝐵𝐶 must be greater, so greater, than the length of the third side. So it must be greater than, in this case, 𝐴𝐶.

So that’s one formulation of the triangle inequality for this triangle. But it says the sum of the lengths of any two sides, which means I can also write this inequality down using the other pairs. So it must also be true that 𝐴𝐵 plus 𝐴𝐶 is greater than 𝐵𝐶. And third, it must also be true that 𝐵𝐶 plus 𝐴𝐶 is greater than 𝐴𝐵. So whichever pair of sides I choose, the sum of those two sides must be greater than the length of the third side.

Now just have to think about why that is the case, that this inequality has to be true. Suppose I asked you to construct a triangle with sides of seven centimeters, four centimeters, and two centimeters. Now to construct means to do so accurately using mathematical equipment, such as a ruler and a pair of compasses. So what you would do if you were attempting to construct this triangle. Well I would start off by accurately drawing out the base side of seven centimeters. And then the way that you would draw accurately this side of four centimeters is you would get your pair of compasses, you would set it to four centimeters, you would place the point at one end of this seven centimeter line, and then you would draw an arc of all the points four centimeters away from that point there. In order to construct the two centimeter side, you’d do the same thing. You’d set your pair of compasses to two centimeters, place the point at this end here, and then you draw an arc of all the points two centimeters away.

Now what you notice is that those arcs don’t cross at any point. So if you were to try and draw your triangle, you would end up with a gap where those lines can’t possibly meet to form the third corner. Now this is why it’s important that the triangle inequality holds true, because all you can see this triangle didn’t work, left a gap. And if you look at those measurements of four, two, and seven, you’ll see that the triangle inequality doesn’t work because I have that four plus two, which is six, well that isn’t greater than seven. And that’s why I wasn’t able to draw this triangle.

So let’s look at a question on this. The first question says: Is it possible to form a triangle with side lengths three inches, five inches, and seven inches?

So what we need to check is does the triangle inequality hold true for all the different pairs of sides here. So if you recall, we’ve got three different inequalities that we need to check. So here’s the first one. If I take the three and the five, is three plus five greater than seven? And of course it is. Eight is greater than seven. So that one works. Now let’s look at the second pair. So let’s take the five and the seven. So is the sum of those two sides greater than the third side? Is five plus seven greater than three? And of course twelve is greater than three, so that one also works. The third one, we need to take the final pair of sides. So we need to look at the sum of the three and the seven. So we’re asking, is three plus seven greater than five? And yes, ten is greater than five. So all three triangle inequalities are satisfied, which means the answer to the question “is it possible to form this triangle”, yes it is. We do of course need the working out in order to back up that answer.

Okay. A very similar question, is it possible to form a triangle with side lengths six meters, seven meters, and eighteen meters? So we know what we need to do from the previous one. We need to check whether these three triangle inequalities hold. But just by casting an eye over these three measurements, six meters, seven meters, and eighteen meters, I can see already that there’s one that’s not going to work. If I look at the sum of the six and the seven, so six plus seven is thirteen, and thirteen is not greater than eighteen. So the sum of this pair of sides does not exceed the length of the third side, which means that, for this question, no it isn’t possible to form such a triangle.

Now an important point to note here, when the answer was yes, we had to check all three of the inequalities and make sure all three of them hold true. When the answer was no, it’s enough to demonstrate that just one pair of sides doesn’t satisfy this inequality. So as long as you can find one of them that doesn’t work, then it won’t be possible to form the triangle. You don’t actually need to check the other two.

Okay. The next question says: Two sides of a triangle are five centimeters and eight centimeters. What is the range of values for the third side?

So we aren’t asked to find the third side explicitly. We’re asked to find the range of possible values that this third side could be. So I don’t know this side, which means I’m gonna start off by allocating a letter. So I’m gonna call it 𝑥. So I’m starting my working out with: Let the length of the third side be 𝑥 centimeters. Now I need to think about those three triangle inequalities that we saw previously. And to start off with, I’m gonna think about the fact that if I take this third side 𝑥 and the side of five centimeters, then the sum of those two sides need to be greater than the eight centimeters. So this gives me my first inequality, and it’s the 𝑥 plus five must be greater than eight. Now to solve this inequality, I need to subtract five from both sides. And this tells me that 𝑥 must be greater than three. So this is my first piece of information about 𝑥; it has to be bigger than three.

Now let’s right down our second inequality. And it’s going to be that if I take this third side 𝑥 and eight centimeters, that has to be bigger than five. So this gives me the inequality 𝑥 plus eight must be greater than five. Now to solve this, I need to subtract eight from both sides. And this tells me that 𝑥 must be greater than negative three. Now that doesn’t really add any extra information because 𝑥 is representing a length. So by definition, it has to be a positive value, which means it must be greater than negative three. So we haven’t really gained anything by using that inequality there.

The third inequality, I need to look at the five and the eight. And the sum of those must be greater than 𝑥. So I have five plus eight is greater than 𝑥. Simplifying that, it tells me that thirteen is greater than 𝑥. Or, if I just write that inequality the other way around, then I have that 𝑥 is less than thirteen.

Okay. Finally, I need to put all of this information together. So that middle inequality, remember, didn’t really tell me anything useful. But the other two do. They tell me, in one case, that 𝑥 is greater than three. And then the second lead, it tells me that 𝑥 is less than thirteen. So if I pull those two inequalities together, it gives me a double-sided inequality for 𝑥. So this tells me that 𝑥 is greater than three but less than thirteen. And so that is the range of possible values for this third side of the triangle.

Now just as an aside here, if the question had asked for the least integer value of 𝑥, for example, then it wouldn’t be three. It would be four, because it has to be strictly greater than three. So we’re looking at the next integer up after three.

So in this question, we formed the three inequalities by looking at the three different pairs of sides of the triangle, solved the inequalities, and then pulled all that information together at the end of the question, in order to work out the range of possible values for this third side.

Okay. The final question, we’re given a diagram and we’re told: Given that 𝑥 is an integer, so a whole number, determine the possible values of 𝑥. And looking at the diagram, we can see that all three sides are expressed in terms of this letter 𝑥.

So just like in the previous example, we have three inequalities that we need to write down. So the first one, I’m gonna take the red side and the blue side and sum them together. And when I do that, the triangle inequality tells me that that must be greater than the third side, the five 𝑥 minus three. So I have this inequality here. Now what I need to do is, I need to solve the inequality. So if I simplify the left-hand side, I’ve got three 𝑥 plus 𝑥 is four 𝑥 and negative one plus three is two. So I have four 𝑥 plus two is greater than five 𝑥 minus three. Next step to solve in this inequality, I want to add three to both sides, which gives me four 𝑥 plus five is greater than five 𝑥. And then finally, subtract four 𝑥 from both sides, which gives me five is greater than 𝑥. Or, if I just turn it around the other way, I have 𝑥 is less than five. So that gives me my first piece of information about 𝑥; it’s that it must be less than five.

Now I need to do the same thing with the other pairs of sides of this triangle. So if I look at the blue side and the green side next, and the sum of those two, so then I have that 𝑥 plus three plus five 𝑥 minus three must be greater than that third side, which is three 𝑥 minus one. Now at the same time, I’m gonna write down my third inequality. So this third one is where I sum the red and the green side, and that must be greater than the blue side. So that gives me this third inequality, three 𝑥 minus one plus five 𝑥 minus three is greater than 𝑥 plus three.

Now I’m not going to go through the step-by-step solution of both of those inequalities. I have written on the screen. So if you want to pause the video and go through it yourself, you can. But it gives us these values here. 𝑥 is greater than negative a third, for the first one, which doesn’t really give us any new information, because the third inequality tells us that 𝑥 has to be greater than one. So if 𝑥 has to be greater than one, it definitely has to be greater than negative a third. And we also need to make sure that the length of these sides is positive. So it’s the first and the last inequalities that give us the most information. 𝑥 has to be greater than one, but it has to be less than five.

So I could write that as a double-sided inequality, as I did before. But this question actually says that 𝑥 is an integer. So 𝑥 is an integer somewhere greater than one but less than five, which means there are three possible integers that 𝑥 could be. 𝑥 could be two, three, or four. So that gives me my answer to this question.

So to summarize then, we’ve seen what the triangle inequality is, and what it tells us about the relationship that must exist between the three sides of a triangle. We’ve seen how to apply it to determine whether it is, or isn’t, possible to create a particular triangle, given three lengths. And then we’ve seen how to apply it to some more complex questions where we’ve had to setup and solve some algebraic inequalities.