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Video: The Pythagorean Theorem - Introduction

Lauren McNaughten

We introduce the term hypotenuse, and explain the exact wording of the Pythagorean theorem and then examine the Perigal's Dissection proof of the theorem. We then see how to use and apply the theorem to show whether a triangle is a right triangle.


Video Transcript

In this video, we are going to be introduced to the Pythagorean theorem, which is a really useful geometric result within mathematics that also has some really interesting applications. Now this theorem is all about right-angled triangles or, more specifically, it’s about the relationship that exists between the lengths of the sides in any right-angled triangle. Its name, the Pythagorean theorem comes from the person that’s named after, so it’s named after a mathematician called Pythagoras who was a Greek mathematician. He was born we don’t know exactly when, but around five hundred and seventy BC, and it’s named after him. Now it may well have been in use before his time, but he is credited with being the first person to actually prove this theorem, and therefore that’s why it’s named after him.

Now before we look at the theorem in detail, we need a little bit of language or terminology that’s associated with right-angled triangles. And what we need to look at is the name that we give to one of these particular sides. And the side that we’re interested in specifically is the longest side of a right-angled triangle. Now in any right-angled triangle the longest side is always a side that is opposite the right angle; so for this triangle here, it’s going to be this side that I’ve marked in orange here. That is the longest side of this right-angled triangle. Now we have a particular name that we give to this side, and the name that we gave it is the hypotenuse of the triangle. So you will see, in here, this word an awful lot when working with right-angled triangles and specifically with the Pythagorean theorem: the hypotenuse of the triangle, meaning the longest side, the side which is opposite the right angle. Now you need to be used to seeing right-angled triangles in lots of different orientations. They won’t always look exactly like the first one I’ve drawn here. But in any case, if you just go opposite the right angle, then you’ll be able to identify easily which side is the hypotenuse.

So now let’s have a look at the statement of the theorem, and I’ve written it out on the screen here. The Pythagorean theorem tells us that in any right-angled triangle — and it must be right-angled; the theorem doesn’t work if we’re not in a right-angled triangle. But what it tells us is the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. So what that means is in any right-angled triangle if I’d look at the length of the hypotenuse whatever it might be and I square it, I will get it exactly the same result as if I square each of the other two sides, which are both shorter than the hypotenuse, and then I add those two squares together. Now it may help you to picture that using a diagram. So the diagram I’ve got here, I’ve got a right-angled triangle and I’ve labelled it’s three sides as 𝑎, 𝑏, and 𝑐. And then I have drawn squares on each of those sides. So for the side where the length is 𝑎, the area of that square is 𝑎 squared; for the side where the length is 𝑏, the area of that is 𝑏 squared; and for the side where the length is 𝑐, the area of that square is 𝑐 squared. And so thinking about it pictorially, what the Pythagorean theorem tells me is it if I add together the area of those two smaller squares, so the red square and the blue square, I will get the area of the largest square, the green square. You will also often see the theorem written in terms of 𝑎, 𝑏, and 𝑐. So here 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared. You will often see the theorem written like that, and it just means adding together those two smaller squares gives me the same result as what I get when I look at the larger square.

Now there’re a lots of different ways that you can convince yourself of the validity of this theorem. For example, you could draw or you could construct a right-angled triangle, and you could measure the length of each of its sides as accurately as possible. And you could test it out: is it true that when you do that you’ll get this result 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared? Now if you do that, you may find you don’t get the results exactly because you’ll be limited by the accuracy of the ruler or whatever you’re using to measure, but you should get something that’s pretty close if the triangle you’ve drawn truly does contain a right angle.

Now another way to see this and in fact a method that you can reproduce for yourself, and it was offered by mathematician called Henry Perigal in eighteen ninety one. And what he suggested was a proof where by you cut up the smaller squares. You say you cut them up in a particular way as illustrated by the lines on this diagram, and then you can actually rearrange the pieces and stick them all directly inside the largest square to show that those two areas do in fact add up to the area of the largest square. And that’s quite a nice physical proof that you can use to illustrate the Pythagorean theorem. Now there are many many other ways that you can see this and other proofs that are in existence. One such proof is a proof called Garfield’s proof, which involves a geometric and algebraic argument to demonstrate this, and that is actually are covered in another video if you’re interested in looking at that. So lots of different ways to prove the theorem, and the theorem itself is really powerful because what we can do is we can use it in a couple of different ways: we can use it, first of all, to test whether or not a triangle is in fact a right-angled triangle, and we’ll see some examples of that in a minute, or we can use it if we know a triangle is right-angled then we can use it in order to calculate the lengths of any of the missing sides as long as we know the other two. And there are some quite interesting applications of that within maths in which we can use the theorem.

So let’s look at testing whether or not a triangle is right-angled by seeing whether or not the Pythagorean theorem holds. So I have a triangle here the sides are five centimetres, twelve centimetres and thirteen centimetres. And the question I’m looking to answer is, “does this triangle contain a right angle?” Now it looks like it does from the way the diagram has been drawn, but we should never assume that just because the triangle looks like it’s right-angled. We can’t make that assumption. So what we need to do instead is to see whether or not the Pythagorean theorem holds for this triangle; and if it does it will be right-angled, if it doesn’t then this triangle will not be right-angled. So a reminder of the Pythagorean theorem, 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared. Now 𝑎 and 𝑏 represented the two shorter sides of this triangle, so that will be the five and the twelve. Now it doesn’t matter which way round I assign 𝑎 and 𝑏; I can do it whichever order. And thirteen represents the hypotenuse for this triangle, so the longest side which is thirteen. So now we need to test whether or not the theorem holds, so I’m going to start off by working out what 𝑎 squared plus 𝑏 squared is equal to for this triangle. So it’s going to be equal to five squared plus twelve squared. Now if I replace both of those with their actual values, then I’ve got twenty-five plus a hundred and forty-four. And if I had those two together, I have a hundred and sixty-nine. Now the numbers in this question of all nice numbers, and I do know that in fact the hundred and sixty-nine is equal to thirteen squared. And thirteen is the length of the third side of my triangle 𝑐, so this is equal to 𝑐 squared. So in this triangle, it does work; five squared plus twelve squared is equal to thirteen squared. So we have 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, and therefore this triangle does contain a right angle. And so this angle here is a right angle.

Now actually this is a particularly special type of right-angled triangle because not only does it contain a right angle, but also all three of its sides are integers: five centimetres, twelve centimetres and thirteen centimetres. And that will not often be the case when you’re answering a question using the Pythagorean theorem. More often than not, you will have side lengths that are decimals. And so this is a particularly interesting type of triangle, and it has a particular name: it’s known as a Pythagorean triple because it is a right-angled triangle where all three of the sides are integers. And you can have a look for other Pythagorean triples to see if you can come up with any more.

Now let’s have a look at another example and see whether we can determine if this triangle is right-angled. So we have a triangle here with sides of length four centimetres, six centimetres, and seven centimetres, and we want to see whether or not the Pythagorean theorem holds true for this triangle. And that will tell us whether or not it contains right angle. So a reminder of the theorem, 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑎 and 𝑏 are the two shorter sides, so I’m going to label this one as 𝑎 and this one is 𝑏, and 𝑐 is the longest side. so That’s the seven centimetres side here. As we did before, let’s start off by calculating the value of 𝑎 squared plus 𝑏 squared. So 𝑎 squared plus 𝑏 squared will be four squared plus six squared. Then if I work out the values of these two quantities, I have sixteen for four squared and thirty-six for six squared. Then I need to add those values together, which gives me a total of fifty-two for 𝑎 squared plus 𝑏 squared. Now 𝑐 is seven centimetres, so I know that 𝑐 squared is equal to seven squared, which is equal to forty-nine. And what I can see is that these two quantities are not equal to each other; on the one hand, I’ve got fifty-two, and on the other hand I’ve got forty-nine. So for this particular triangle, 𝑎 squared plus 𝑏 squared is not equal to 𝑐 squared and therefore isn’t a right-angled triangle.

Now this is an illustration of the converse of the Pythagorean theorem, so the opposite, which tells us that if 𝑎 squared plus 𝑏 squared is not the same as 𝑐 squared, then the triangle is not a right-angled triangle. So it doesn’t matter how much a triangle may look like a right-angled triangle from a diagram you’ve been given; if the Pythagorean theorem doesn’t work for the three sides, then it isn’t a right-angled triangle. So there you have it, an introduction to the Pythagorean theorem, some examples of how you can convince yourself of its validity, and also some examples of how you can use it to test whether or not a triangle truly is a right-angle triangle.