Video: The Pythagorean Theorem

In this video, we will learn how to use the Pythagorean theorem to find the length of the hypotenuse or a leg of a right triangle and its area.

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Video Transcript

In this video, we will learn how to use the Pythagorean theorem, or Pythagoras’ theorem as you may know it, to find the lengths of sides in right triangles. This is hugely useful as right triangles can be used to model a lot of different physical scenarios. And they’re often involved in problems involving area. The Pythagorean theorem is one of those really well-known bits of maths. And many people will still remember its name from their school days even if they can no longer remember what the theorem itself actually says.

So, what does the Pythagorean theorem say? Well, the Pythagorean theorem is all about the special relationship that exists between the lengths of the three sides in a right triangle, that is, a triangle that includes a right angle. Remember, we call the longest side of a right triangle, which is always the side directly opposite the right angle, the hypotenuse. The Pythagorean theorem then says this.

In a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. Often, we use the letters 𝑎 and 𝑏 to represent the two shorter sides or legs of the right triangle. And we use the letter 𝑐 to represent the hypotenuse, in which case the Pythagorean theorem can be expressed as 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared. But it’s important to remember what the theorem itself is saying, not just to learn this equation.

In pictorial terms, what the Pythagorean theorem is telling us is that if were to draw a square on each side of a right triangle, then the sum of the areas of two smaller squares would equal the area of the largest square. That’s the square on the hypotenuse. There are lots of different ways to prove the Pythagorean theorem, but one of the nicest, in my opinion, is a method called Perigal’s dissection. We won’t go into detail here. But it involves chopping the two smaller squares up and rearranging the pieces to fit exactly inside the larger square, as you can see in the diagram here. If you like, you could try this yourself by reproducing this diagram on a piece of paper.

Let’s now have a look at some examples of how we can apply the Pythagorean theorem. We’ll begin by considering example of how to use the theorem to find the hypotenuse of a right triangle.

Find 𝑥 in the right triangle shown.

Looking at the information we’ve been given, we note, first of all, that this triangle is a right triangle. It includes a right angle. And we’ve been given the lengths of two of its sides. They are eight units and 15 units. 𝑥 represents the length of the third side of this right triangle. And from its position, directly opposite the right angle, we note that 𝑥 is the hypotenuse of this triangle. As we’ve been given the lengths of two sides in a right triangle and we wish to calculate the third, this is exactly the setup we need in order to apply the Pythagorean theorem.

This tells us that, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. So we’ll begin by writing down what the Pythagorean theorem tells us about this triangle in particular. The two shorter sides are eight units and 15 units. So the sum of the squares of the two shorter sides is eight squared plus 15 squared. This is then equal to the square of the hypotenuse. And as the hypotenuse of our triangle is 𝑥, we now have the equation eight squared plus 15 squared is equal to 𝑥 squared.

So by considering what the Pythagorean theorem tells us about this triangle in particular, we have an equation we can solve in order to determine the value of 𝑥. You may prefer to swap the two sides of the equation around so that 𝑥 is on the left-hand side, although this isn’t entirely necessary. Now that we formed our equation, we’re going to solve it by first evaluating eight squared and 15 squared. This gives 𝑥 squared equals 64 plus 225, which simplifies to 𝑥 squared equals 289.

The next step in solving this equation is to take the square root of each side because the square root of 𝑥 squared will give 𝑥. Now usually, when we solve an equation by square rooting, we must remember to take plus or minus the square root. But here 𝑥 has a physical meaning; it’s the length of a side in a triangle. So it must take a positive value. We therefore write 𝑥 equals just the positive square root of 289. 289 is in fact a square number, and its square root is 17. So we found the value of 𝑥. 𝑥 is equal to 17.

Now, we should always perform a quick sense check of our answer by comparing the value we found with the other two sides in the triangle. Remember, 𝑥 represents the hypotenuse, which is the longest side in this right triangle. So our value for 𝑥 needs to be bigger than the lengths of the two other sides. Our value is 17 and the two other sides are 15 and eight. So our answer does make sense.

Now, in fact, this triangle is an example of a special type of right triangle, called a Pythagorean triple. This is a right triangle in which all three of the side lengths are integers. The most well-known Pythagorean triple is the three-four-five triangle as three squared plus four squared is equal to five squared. You may well encounter Pythagorean triples when working without a calculator. So it’s a good idea to be familiar with some of the most common ones. By applying the Pythagorean theorem then, we’ve found the value of 𝑥 in the right triangle shown is 17.

In our next example, we’ll see how to apply the Pythagorean theorem to find the length of one of the two shorter sides in a right triangle.

Find 𝑥 in the right triangle shown.

So we have a right triangle. And we’re asked to find the value of 𝑥, which represents the length of one of the triangle sides. We’ve been given the lengths of the other two sides. So we have exactly the right set of information in order to apply the Pythagorean theorem. This tells us that, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. Now, before applying the Pythagorean theorem, we must be very careful to make sure we correctly identify which side of the triangle is the hypotenuse. And remember, it’s always the side directly opposite the right angle. So in this case, the hypotenuse of the triangle is 13 units.

The side we’ve been asked to find, length 𝑥, is one of the two shorter sides or legs of this right triangle. So the first thing we do then is to write down what the Pythagorean theorem tells us about this particular triangle. The two shorter sides are 𝑥 and 12. So the sum of their squares will be 𝑥 squared plus 12 squared. The hypotenuse of the triangle is 13 units. So if the sum of the squares of the two shorter sides is equal to the square of the hypotenuse, we have the equation 𝑥 squared plus 12 squared equals 13 squared.

Now that we’ve formed our equation, we’re going to solve it to determine the value of 𝑥. First, we evaluate 12 squared and 13 squared, giving 𝑥 squared plus 144 equals 169. We want to leave 𝑥 or 𝑥 squared initially on its own on the left-hand side of the equation. So the next step is to subtract 144 from each side. On the left-hand side, 𝑥 squared plus 144 minus 144 just leaves 𝑥 squared. And on the right-hand side, 169 minus 144 is 25.

The final step is to take the square root of each side of the equation, remembering we only need to take the positive square root as 𝑥 represents a length. So it must have a positive value. 𝑥 is therefore equal to the square root of 25. And as 25 is a square number, its square root is an integer; it’s simply five. So we found the value of 𝑥. 𝑥 is equal to five. Now, in fact, this triangle is an example of a Pythagorean triple. That’s a right triangle in which all three side lengths are integers.

We should also perform a quick check of our answer. Remember, we were looking to calculate one of the shorter sides of this triangle. So our value for 𝑥 must be less than the length we were given for the hypotenuse. Five is certainly less than 13. So our answer makes sense. So by applying the Pythagorean theorem, we’ve solved this problem. The value of 𝑥 is five. We must make sure we’re really careful when setting up our equation. And we need to be sure before we begin whether we’ve been asked to find the length of one of the shorter sides or the length of the hypotenuse.

So we’ve now seen one example of calculating the length of the hypotenuse and one example of calculating the length of one of the shorter sides. The Pythagorean theorem is really useful because it helps us answer lots of different types of practical problems. So we’ll now consider a couple of examples with a greater focus on problem solving.

Determine the diagonal length of the rectangle whose length is 48 centimetres and width is 20 centimetres.

Now, we haven’t been given a diagram for this question. So it’s always a good idea to begin by drawing our own. We have a rectangle with a length of 48 centimetres and a width of 20 centimetres. The length we’ve been asked to calculate is the diagonal of this rectangle. That’s the line that joins opposite corners together. We can use the letter 𝑑 to represent this unknown length. Now we know that all the interior angles in a rectangle are 90 degrees. So, in fact, this problem isn’t just about rectangles. It’s also about right triangles, that is, the triangle formed by the rectangle’s length, its width, and this diagonal.

Looking at the lower triangle in our diagram, we can see that we’ve been given the lengths of two of its sides — they’re 20 centimetres and 48 centimetres — and asked to calculate the length of its third side. And as this is a right triangle, we’re going to be able to do this by applying the Pythagorean theorem. This tells us that, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. Now, before we try to apply the Pythagorean theorem, we must identify which of the three sides we’ve been asked to calculate. Remember, the hypotenuse is always the side directly opposite the right angle. So the side we’re looking to find is the hypotenuse of a triangle.

We then ask ourselves, “what does the Pythagorean theorem tell us, not just in general, but about this triangle specifically?” Well, as the two shorter sides are 48 and 20 centimetres, the sum of their squares is 48 squared plus 20 squared. And the square of the hypotenuse is 𝑑 squared. So we have the equation 48 squared plus 20 squared equals 𝑑 squared. We can of course swap the two sides of the equation round if we prefer to have 𝑑 squared on the left-hand side. So by applying the Pythagorean theorem, we’ve formed an equation, which we can now solve in order to determine the value of 𝑑.

First, we evaluate 48 squared and 20 squared and then add these values together to give 𝑑 squared is equal to 2704. The final step in solving this equation is to take the square root of each side, giving 𝑑 equals the square root of 2704. Now, 2704 is in fact a square number although probably not one that you are overly familiar with. Its square root is simply 52. So we have that 𝑑 is equal to 52. The diagonal length of this rectangle then is 52 centimetres.

Now, we should perform a quick sense check of our answer. Remember, 𝑑 was the hypotenuse of this triangle. It’s supposed to be the longest side. So we need to check that our value does make sense. Well, 52 is indeed greater than each of the other side lengths. So it’s a sensible value for the hypotenuse of this triangle. So we’ve completed the problem. The key stage in this question was to first draw our own diagram. And once we did, we saw that this problem wasn’t just about rectangles. It was in fact about right triangles. And hence, we could solve it by applying the Pythagorean theorem.

Let’s now consider one final example involving points plotted on a coordinate grid.

A triangle has vertices of the points 𝐴 four, one; 𝐵 six, two; and 𝐶 two, five. Work out the lengths of the sides of the triangle. Give your answers as surds in their simplest form. And secondly, is this triangle a right triangle?

Let’s begin by sketching this triangle on a coordinate grid. We absolutely don’t need to plot this triangle accurately. We aren’t going to be measuring the lengths of any of the lines. We just want to sketch it using the approximate position of these three points relative to one another.

So the triangle looks a little something like this. Now, from our sketch, it looks possible that this could be a right triangle with the right angle at 𝐴. But we can’t confirm this from our sketch. Let’s consider the first part of the question. We need to find the lengths of the three sides of the triangle. And we’ll begin by finding the length of the side 𝐴𝐵.

We can sketch in a right triangle below this line using 𝐴𝐵 as its hypotenuse. We can also work out the lengths of the other two sides in this triangle. The horizontal side will be the difference between the 𝑥-values at its endpoints. That’s the difference between six and four, which is two. And the vertical side will be the difference between the 𝑦-values at its endpoints. That’s the difference between two and one, which is one.

As we now have the lengths of two sides in a right triangle and we wish to calculate the length of the third side, we can apply the Pythagorean theorem, which tells us that, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. Remember, 𝐴𝐵 is the hypotenuse. So we have that 𝐴𝐵 squared is equal to one squared plus two squared. One squared is one and two squared is four. So adding these values together, we have that 𝐴𝐵 squared is equal to five.

To find the length of 𝐴𝐵, we need to square root each side of this equation. And remember at this point, we’ve been told to give our answer as a surd. So we have that 𝐴𝐵 is equal to root five. We can find the lengths of the other two sides of the triangle in the same way. We sketch in a right triangle below the line 𝐵𝐶. And we see that it has a horizontal side of four units and a vertical side of three units.

𝐵𝐶 is the hypotenuse of this triangle. So applying the Pythagorean theorem, we have that 𝐵𝐶 squared is equal to three squared plus four squared. That’s nine plus 16, which is equal to 25. 𝐵𝐶 is therefore equal to the square root of 25, which is simply the integer five. In the same way, 𝐴𝐶 is the hypotenuse of a right triangle with shorter sides of two and four units. So 𝐴𝐶 is equal to the square root of 20, which simplifies to two root five.

So we’ve answered the first part of the question. And now we need to determine whether this triangle is a right triangle. Well, if it is, then the Pythagorean theorem will hold for its three side lengths. Now we suspect it that the right angle was at 𝐴, which would make 𝐵𝐶 the hypotenuse of the triangle if it is indeed a right triangle.

We therefore want to know whether 𝐵𝐶 squared is equal to 𝐴𝐵 squared plus 𝐴𝐶 squared. Well, we can in fact use the squared side lengths. We know that 𝐵𝐶 squared is 25. We know that 𝐴𝐵 squared is five. And we know that 𝐴𝐶 squared is 20. So is it true that 25 is equal to five plus 20? Yes, of course, it’s true, which means that the Pythagorean theorem holds for this triangle. And therefore, it is indeed a right triangle. So we’ve completed the problem. We have the three side lengths. 𝐴𝐵 equals root five, 𝐵𝐶 equals five, and 𝐴𝐶 equals two root five. And we’ve determined that the triangle is a right triangle.

Let’s now summarise what we’ve seen in this video. The Pythagorean theorem tells us that, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse, which we may often see written as 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared. The first step in any problem should be to write down what the Pythagorean theorem tells us about the particular triangle in this problem. That is, we form an equation. We then solve our equation, which will involve square rooting. Finally, we should always check our answer by making sure that the value we’ve calculated makes sense in relation to the lengths of the other two sides.

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