Lesson video

Hello, I'm Mrs. Lashley, and I'm going to be working with you as we go through this lesson today.

I hope you're ready to try your best and give it all a go as best as you can and I'm gonna help you along the way.

So, our learning outcome today is to be able to use Pythagoras's theorem to calculate the length of a side of a right-angled triangle.

Some key words that are associated with today's lesson are the hypotenuse, and the hypotenuse is the side of a right-angled triangle which is opposite the right angle.

It's the longest edge of a right-angled triangle.

And then Pythagoras's theorem states that the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of the hypotenuse.

And we just learned that the hypotenuse, or we just reminded ourselves, that the hypotenuse is that longest edge opposite the right angle.

So, for this lesson on checking and further securing understanding of Pythagoras's theorem, we're going to split the lesson into three learning cycles.

The first learning cycle is looking at finding the length of the hypotenuse, so that longest side, the side that is always opposite the right angle.

Then we'll move on to the second learning cycle where we'll look at using Pythagoras's theorem to find the length of a shorter side of a right-angled triangle.

And then finally, we'll bring that all together to work out the length of any side of a right-angled triangle.

Let's make a start at reminding ourselves of how we find the hypotenuse using Pythagoras's theorem.

So, the figure below shows the triangle ABC.

We can see that the vertices are marked as A, B, and C.

Which side of that triangle is the longest? Have a moment to think about that.

Okay, so which side is the shortest? So, how can you be sure that your answers to those two previous questions are correct, despite the fact that you do not know any lengths of that right-angle triangle? Well, the longest side is AB.

The shortest side is AC.

So, what was your justification for knowing that those two edges were the longest, and shortest sides of this right-angle triangle? Well, the longest side of the triangle is opposite the largest angle.

And if there is a right angle, if there is a 90-degree angle in the triangle, then that will be the largest angle because there would only be 90 degrees left, and that has to be split between two angles.

So, 90 degrees or a right-angle triangle, then that is the largest angle, and therefore the opposite edge is the longest side.

We call that the hypotenuse.

The shortest side is therefore opposite the smallest angle.

And in this case, we had the labels on the angles.

We knew that one was 30 degrees, and therefore AC was the shortest side.

So, the longest side of a right-angle triangle is called the hypotenuse.

That was one of our key words, and we've used it already this lesson.

So, the hypotenuse is here on this diagram because it is opposite the right angle.

It will always be opposite the right angle in a right-angle triangle.

Pythagoras's theorem states that the sum of the squares of the two shorter sides of a right-angle triangle is equal to the square of its longest side, which is the hypotenuse.

So, when we're talking about squares, we're talking about squaring the length, and if you square the length, you're actually finding the area of a square.

So, here this diagram you're probably familiar with it, shows Pythagoras's theorem with this idea of squares.

So, the two smallest squares that come off of the shorter edges, the sum of those two areas is equal to the area of the square on the longest side, which we call the hypotenuse.

It can also be expressed as a squared plus b squared equals c squared, or c squared equals a squared plus b squared, where c is the hypotenuse.

So, we can use Pythagoras's theorem to find the length of the hypotenuse of a triangle.

So, here we've got a triangle with two shorter edges of three centimetres and four centimetres.

Method one is to manipulate the diagram, and the diagram is the one with the squares on each edge.

So, if you have this diagram, we can work out the areas of the squares.

A square with an edge of three centimetres will have an area of nine square centimetres.

The square with an edge of four centimetres will therefore have an area of four squared, which is 16.

So, Pythagoras's theorem tells us that the sum of those two squares is equal to the square of the longest edge.

So, the sum of 9 and 16 is 25.

And so now we have to think, okay, we have a square with an area of 25.

What would the edge length be? And we can use square rooting to find that edge length, which is five centimetres.

So, we have calculated the length of the hypotenuse of this particular right-angle triangle.

The second method is to not worry about the diagram, but instead to use the formula of Pythagoras's theorem.

So, a squared plus b squared equals c squared, where c is the hypotenuse.

So, on your triangle, you need to identify where the hypotenuse is, always opposite the right angle, and then you need to think of the shorter sides as a and b.

It doesn't matter which one you decide is a, and which one is b.

You can see here that I've said that three centimetres is going to be a, and four centimetres is going to be b.

So, the next step is to substitute into the formula.

So, three squared plus four squared equals c squared.

Now we're going to evaluate the squares.

So, three squared is nine, and four squared is 16, and then we are going to add them together.

We're going to find the sum of those two squares, which is 25.

We now know that c squared equals 25.

So, what value of c, when squared, gives you 25? Well, that would be five.

So, triangles can be in different orientations, but the right angle can be used to find which side is the hypotenuse.

So, whenever you find yourself with a right-angle triangle, it's very easy to identify the hypotenuse because it's the longest edge opposite the largest angle.

So, on this diagram here, this edge that we do not have a length for at this moment in time is the hypotenuse.

So, we can then use Pythagoras's theorem: a squared plus b squared equals c squared.

Here we've got an isosceles triangle, because you can have right-angled isosceles triangles.

We substitute in those shorter edges, evaluate the squares, find the sum.

Now we know that c squared is equal to 648.

So, we can use our calculator to square root 648, and it gives our answer in a simplified surd form: 18 root 2.

Sometimes you may need to change the format of your answer so that the hypotenuse is a length as a decimal, for example.

The question will state how to leave your answer normally.

So, on a calculator, you might need to change the format, and you can do this by pressing format, and then moving down to the decimal, and then pressing OK.

And now we can see that 18 root 2 is approximately equivalent to 25.

45584412, and decimals may therefore need to be rounded to an appropriate degree of accuracy.

So, for this one here we might write 25.

5 to one decimal place.

So, here's a check for you.

Calculate the unknown length in this triangle, which is the hypotenuse, and give your answer in decimal form, accurate to one decimal place.

So, pause the video, and then when you're ready to check, press play.

So, we're gonna use the formula, or potentially use the diagram, but substituting in the shorter edges, 8 squared plus 15 squared is 289.

So, 289 is equivalent to the hypotenuse squared, and if we square root 289, we get 17.

So, the hypotenuse is 17.

0 centimetres to one decimal place.

Because the question stated that you needed to give it accurate to one decimal place, then the point zero is necessary in your answer.

So, we're on to the first task of this lesson, and that is to calculate the unknown lengths and round your answers to one place where appropriate.

So, there are six different triangles.

You need to work out the hypotenuse in each case by using Pythagoras's theorem.

Press pause, and when you're ready for the answers, press play.

So, we're going to go through the answers to the six triangles here.

So, the first one is an integer value; you didn't need to round it: 10 centimetres.

B is also an integer value: 26 centimetres.

C: 25 centimetres.

On D you did need to round it to one decimal place, so 47.

2, and this time the units are millimetres.

E: 6.

0 metres, and F was an isosceles triangle; you knew that from the hash marks, and so the hypotenuse would be 5.

7 centimetres to one decimal place.

So, we're now gonna move to the second learning cycle, where we're gonna use Pythagoras's theorem, but this time to find the short side.

So, Pythagoras's theorem can be used to find the length of one of the shorter sides of a right-angled triangle.

So here you can see we have the hypotenuse, that's the one opposite that right angle, and it has a length of 10 centimetres.

One of the other short sides is 6 centimetres, and we're required to find the third edge length.

So, method one once again is manipulating the diagram.

If we draw the squares on each edge of our triangle, we know that the areas of the two smaller squares sum to the area of the larger square that comes off of the hypotenuse.

We can use the length of the shorter edge, 6 centimetres, to work out the area of a square with an edge of 6, which would be 36.

We can do the same for the hypotenuse, and that would have an area of 100.

If the smaller two squares sum to be the larger square, then that third square must have an area of the difference between 100 and 36, which is 64.

So, we've done a subtraction to work out what that area must be.

So now we have a square with an area of 64 square centimetres.

What would the edge length be? Well, we can square root to find that the edge length would be 8 centimetres.

So, this way with using the diagram is one way that we could do it.

Alternatively, we could use the formula.

So, a squared plus b squared equals c squared is Pythagoras's theorem, where c is the hypotenuse.

So once again, we need to identify our edges.

We know that 10 is the hypotenuse.

It is opposite the right angle, and a and b can be either way around.

We're gonna substitute in our values, evaluate the squares, and then we're going to work out what b squared is by rearranging our equation.

B squared is equal to 64, and therefore b equals 8 with square rooted.

So here is a check for you.

Calculate the unknown length in this triangle.

Give your answer in decimal form, accurate to one decimal place.

So, pause the video, and when you're ready to check, press play.

37 was the hypotenuse.

It was opposite the right angle, and 12 was one of the shorter edges.

To work out the area of a square that would be along that unknown edge, we would do the area of the square on the hypotenuse minus the area of the square on the shorter side.

And then we can square root it to get our edge length, which would be 35.

So, the missing length is 35.

0 centimetres.

Remember if it asks you for one decimal place, then you must give your answer with one decimal place.

So, task b.

Calculate the unknown lengths and round your answers to one decimal place where appropriate.

So, pause the video, and then, when you're ready for the answers to this question and this task, press play.

So, question part a: The missing length, the short side, was 6 centimetres.

On b, it was 24 millimetres.

On c, it was 8 centimetres.

So, so far all of those are integers, and therefore they are all Pythagorean triplets or triples.

D: The answer was 1.

7 units to one decimal place.

E: 11.

6 metres.

So, check what units you are in.

And on f we've got an isosceles triangle, and so we need to work out the length of the equal edges, and they would be 14.

1 inches each to one decimal place.

So finally, we're up to the third learning cycle of today's lesson, where we're going to bring it all together and find the length of any side of a right-angle triangle using Pythagoras's theorem.

So, the calculations you perform when using Pythagoras's theorem may differ depending on which length on the triangle you are finding.

And we saw that in the first and second learning cycle.

So, if we were looking to find the hypotenuse, if we look at the left-hand side, we know it's the hypotenuse we're trying to find because that edge is opposite the right angle.

So how do we go about it? Well, we're going to square the edges, add them to find the sum of the two shorter side squares, and then we'll square root to get the edge length, which is the hypotenuse.

So, in this case it's 9.

4 to one decimal place.

If we were finding one of the shorter sides on the right-hand side, so the hypotenuse is 8 centimetres and our shorter edge is 5 centimetres, then what do we do? Well again, we're going to square our edge lengths, and that's to work out the area of a square with the same edge length.

But this time we're looking at the difference because Pythagoras's theorem states that the sum of the two shorter squares is equal to the larger square, and so we need to find the difference because we have the area of the larger square.

So, the difference is 39, and then we would square root it to find the edge length, and that is 6.

2 to one decimal place.

So, at which point do the calculations differ? So just pause the video and have a look.

Run back through those stages of the work, and at which point do they change? Press play when you're ready to move on.

So, the difference is whether we add or whether we subtract.

So, the sum of the two shorter ones gives you the hypotenuse, whereas if you have the hypotenuse then you'll need to find the difference to find a missing short side square.

The way that the formula c squared equals a squared plus b squared is used may differ depending on which length of the triangle you are finding.

So once again, let's go through a hypotenuse style question and a short side.

So, if we're trying to find the hypotenuse using the formula, then we would substitute in our shorter edges that we know, evaluate our squares, find the sum, and then square root to get c, in this case which is the hypotenuse.

If we're trying to find the shorter side, we start with the formula, we substitute in our values in the appropriate terms and then we evaluate our squares, we rearrange it and then we square root.

Our answer is 6.

2 to one decimal place.

So once again at which point do the solutions begin to differ? Obviously, the final answer is different, we're calculating different things, but at which point do the solutions begin to differ.

Pause the video, and when you're ready to check, press play.

So, they begin to differ right at the start, so where we substitute the values because on the right-hand side, we have the c value, we have the hypotenuse, whereas on the left-hand side, we don't have the c value, that is what we are trying to calculate.

So right at the beginning is when things will change, and then that obviously follows through as you go down your solution.

So here is a check, which calculation or calculations may be used as part of the solution for finding the missing length in this triangle? Pause the video, and when you're ready to check press play.

So, we would use c, we have the value of the hypotenuse, it's 15, it's opposite the right angle, and we have a shorter edge.

D has also got a subtraction, but 7 squared is less than 15 squared, and therefore we're gonna find ourselves with a negative value, and you can't have a negative area.

So be mindful when we are doing subtraction that you are subtracting in the correct way.

So, what's about this one? Which calculation or calculations may be used as part of the solution for finding the missing length on this triangle? Once again, pause the video, and when you're ready to check, press play.

So, we could use the first or the second a or b and that's because when we think about the shorter sides as a and b, it doesn't matter which one you call a and which one you call b.

So, 15 squared plus 7 squared is probably where you made 15 a, and 7 b, whereas 7 squared plus 15 squared is probably where you made a equal to 7 and b equal to 15.

Either way, you're going to get the same result.

So, we're up to the last task of today's lesson, and you need to calculate the unknown lengths and round your answers to two decimal places where appropriate.

So, there's eight triangles, remember, they could be the hypotenuse you are trying to find, or it might be a shorter side.

Press pause, and when you're ready to go through the answers to this task, press play.

So, on a it was the hypotenuse, and it's 15 centimetres.

Remember, the hypotenuse is the longest edge.

So, if you were trying to calculate the hypotenuse and your answer is less than the other two edges, then it cannot be correct because the hypotenuse, by definition, is the longest edge of a right-angled triangle.

B, you are not working out the hypotenuse, so our answer should be less than 12 because the hypotenuse is 12 and your answer should have been 7.

94 to two decimal places.

On c, once again, it's a shorter edge, so it should be less than six centimetres the given hypotenuse.

It should be 4.

47 centimetres to two decimal places.

On d, it is the hypotenuse; it's the edge opposite the right angle, so it should be greater than three or one, so 3.

16 metres.

The units were in metres this time.

On to e, it's an isosceles triangle.

We know it's an isosceles triangle because of that 45-degree angle that is marked as well, and if we calculated the third angle, it would have to be 45 degrees in order for the interior angles of the triangle to sum to 180.

So, you should have worked this out as 1.

41 metres to two decimal places.

On f, once again it's another isosceles triangle, and you needed to note that in order to be able to use Pythagoras's theorem.

It's the hypotenuse you're trying to calculate this time, so it should be greater than two; it's 2.

83 metres to two decimal places.

G another isosceles triangle, so to calculate the hypotenuse, 2.

83 metres.

F and g are the same question it's just the other edge of the isosceles that was labelled.

And on to h the answer is one metres.

With h, did you spot a quicker method? Did you need to use Pythagoras's theorem? Well, no, because if we think of two copies of this back-to-back congruent triangles, then that top angle would be a 60-degree angle, and we can see that both of those edges are two metres, and what triangles have got 60 degrees and at least two equal edges? Well, an equilateral triangle, which means that the base edge would also be two, and we only want half of that because we did two copies of this triangle, so therefore the base length would be one metre.

So, to summarise today's lesson on checking and further securing understanding of Pythagoras's theorem, the sum of the squares of the two shorter sides of a right-angle triangle is equal to the square of the longest side.

The longest side of a right-angle triangle is always opposite the right angle, and it's always called the hypotenuse.

The formula a squared plus b squared equals c squared can be used to find the length of the hypotenuse, or it can be rearranged to find the length of one of the shorter sides, and if a right-angle triangle is not immediately available, try to construct one within the problem.

Well done today, and I look forward to working with you again in the future.