Lesson video

Hello, my name is Dr.

Rolson and I am happy to be helping you with your learning during today's lesson.

Let's get started.

Welcome to today's lesson from the unit of trigonometry.

This lesson is called checking and securing understanding of Pythagoras' theorem.

And by the end of today's lesson we'll be able to use Pythagoras' theorem to find the length of any side on a right angle triangle.

Here are two previous keywords that we're going to use again during today's lesson, so you may want to pause the video if you need to remind yourselves what any of these words mean before pressing and play To continue.

This lesson contains four learn cycles where we'll be practising using Pythagoras' theorem in a different way each time.

And let's start by find the length of the hypotenuse.

Here we have a right angle triangle ABC.

Now we don't know the length of any of the sides of this triangle, but which side is the longest? Which side is the shortest? And how can we be sure without knowing the lengths of any of the sides? Pause the video while you think about these questions and then press play when you're ready to continue.

The longest side in this triangle is side AB.

The shortest side is side CA, but how can we be sure without knowing any of the lengths? Well, it's all to do with the angles.

The longest side on a triangle is always opposite the largest angle.

In this case, the largest angle is the right angle, and the side which is opposite that one is side AB.

So AB must be the longest and the shortest side on a triangle is always opposite the smallest angle.

In this case, the smallest angle is 30 degrees, and the side which is opposite is side CA, so that one must be the shortest.

The longest side in a right angle triangle is called the hypotenuse.

So in this triangle, this side would be the hypotenuse, and the hypotenuse is always opposite the right angle.

That's because the longest side is always opposite the largest angle, and in a right angle triangle, the right angle is always the largest angle.

Pythagoras' 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 longer sides, the hypotenuse.

So in this case, if we have lengths a, b and c, where c is the length of the hypotenuse, if I draw a square coming off each of those three sides, it would look something like this and the areas of each of those squares would be a squared, b squared and c squared.

We can see then that the areas of the two smaller squares would add up to the area of the larger square.

Thinking algebraically about this, Pythagoras' theorem can be expressed as a formula as c squared equals a squared plus b squared, where c is the hypotenuse of the triangle.

Now, pyrosis serum can be used to find a length of a hypotenuse on a right triangle such as this one.

We have lengths six centimetres and eight centimetres, which are the two shorter sides of this triangle, and we wanna find the length of the longer side, the hypotheses.

Now there's a couple of ways you can use Pythagoras with this.

One way could be by manipulating the diagram.

We could draw a square off each of these sides like this.

We could then find the areas of the two smaller squares to get 36 centimetre squared and 64 centimetres squared.

We could add those two areas together to get 100 centimetres squared, and then if that's the area of the square, to find the length of that square, we could square root 100 to find the length of 10 centimetres.

Another method could be to use the formula that is a squared plus b squared equals c squared.

So we would label the size of this triangle a, b, and c.

Now c has to be the hypotenuse, but it doesn't really matter which way around a and b go because we're gonna be squaring them and then adding them so it doesn't really matter which way round they go.

So let's substitute the numbers in.

We get six squared plus eight squared equals c squared.

We can then work out the left hand side of this equation now to get 36 plus 64 equals c squared and then 100 equals c squared.

Then to solve this equation, we would square root both sides to get c as 10.

So length is 10 centimetres.

Now one thing that can make using Pythagoras seem a little bit tricky, can be but the triangles can be in different orientations.

But it's not as tricky as we think because the right angle can be used to find which side is the hypotenuse.

The hypotenuse is always opposite the right angle.

So in this case, if I was trying to find the missing length on this triangle, I'll be trying to find the hypotenuse.

I could substitute the numbers I have into the formula version of Pythagoras' theorem, simplify my equation and then when I get to the point that I have c squared equals 648, I'm gonna have another little problem now, that 648 is not a square number.

So when I square root it on my calculator, it may come up with something like this.

To get past this to turn it into a decimal, I can just change how it's presented on the calculator.

On this calculator, I would do it by pressing format.

I'll press the down key until I've highlighted decimal and then press okay.

And now we can see that the length is 25.

45584412.

Now there are more digits, just that's all the calculators showing.

We may need to round our answer to an appropriate degree of accuracy.

In this case, one decimal place would be 25.

5.

So let's check what we've learned.

Here we have a right angle triangle, find the missing length and give your answer in decimal form accurate to one decimal place.

Pause a video while you do this and then press play when you're ready for an answer.

Well, with whatever way you do this, you'll find yourself squaring 12 and 20, adding them together and then square root in it, and your answer will be 23.

3 centimetres.

Okay, it's O2 now for task A.

This task contains one question and here it is.

You have six right angle triangles and you need to find the missing lens, for any decimal answers you get round them to one decimal place.

Pause the video while you have a go and press play when you're ready for answers.

Let's now go through some answers.

Part a, it's 13 centimetres, in part b, five centimetres, in part c, 3.

3 centimetres, part d, 68.

6 millimetres, in part e, 5.

7 metres, and part f, we only have one of the lengths, but we can see that the two short sides are equal to each other because we have that marking on the edges.

So our missing length is six inches and 8.

5 inches.

So far, so good.

Now let's practise finding a short length on a right angle triangle.

Pythagoras' theorem can be used to find a length of one of the shorter sides on a right angle triangle like this one here.

The hypotenuse is 10 centimetres, we have one of the short lens is six centimetres, and we wanna find the third edge on this triangle.

Once again, we can use Pythagoras' theorem and different ways to do this, one way being manipulating the diagram.

We could draw a square off each of the sides as right angle triangle, find the areas of the squares where we know the length, that's 36 centimetres squared, 100 centimetres squared.

Now remember the areas of the two smaller squares add up to the area of the bigger square.

So that means to find the area of one of those smaller squares, I'll need to use subtraction.

I need to do 100 subtract 36 to find the area of the bottom square that is 64 centimetres squared, and then to find the length of that one, I would square root it to get eight.

The other way I could do it is by substitute the numbers I know into the formula version of Pythagoras' theorem.

So a squared plus b squared equals c squared.

I would label the sides of my triangle.

Now c is the longer side, the hypotenuse, so that one has to be 10.

Now it doesn't really matter which one is labelled a and b, but I've labelled the one I know as a and I unknown as b.

Substitute these numbers into my format to create an equation and then simplify my equation and rearrange it to find the value of b which is eight.

Check what we've learned.

Find the missing length in this triangle and give your answer in decimal form to one decimal place.

Pause the video while you work this out and press play when you're ready for an answer.

Well, whichever way you do this, when you use Pythagoras' theorem, you'll find yourself squaring 22 and squaring 12 and then subtracting your answers to get 340 and then square rooting that to get 18.

4 centimetres to one decimal place.

It's over to you again now for task B.

This task contains one question and here it is.

Once again you've got six triangles and you need to find a missing lengths, and give your answer as decimals to one decimal place.

Pause video while you do this and press play when you're ready to continue.

Let's see how we got on with that.

Part a, the missing length is eight centimetres, part b, 10 millimetres, part c, 15 centimetres, part d, it's 0.

6 units, part e it's 9.

8 centimetres and part f, once again we only know one of the lengths, which is the hypotenuse in this case, but we know that the two short aside have the same length because those markings on them.

So when we use Pythagoras' theorem here to work out the shorter sides, we get 4.

2 inches each time.

So if you can find the length of the hypotenuse and you can find the length of a shorter side, let's see now if we can choose the right approach for the right situation.

The calculations you perform when your using Pythagoras' theorem may differ depending on which length on the triangle you're trying to find.

For example, here we have two questions.

They're both right angle triangles and they both have length five centimetres and eight centimetres.

But the question on the left, we're finding in the hypotenuse and with the question on the right, we're finding one of the shorter sides.

Let's see how our methods and calculations may differ.

If we perform each calculation separately.

For example, if you're drawing the squares off of the sides, then this is what your calculations may look like for the left hand example, and this is what your calculations may look like for the right hand example.

Now these calculations look very, very similar, but at what point do they differ? Pause the video while you think about this and press play when you're ready to continue.

These calculations look very, very much the same until we get to this point here.

When we're finding the hypotenuse, we are adding together the squares of the shorter sides, but when we are finding on the shorter sides, we perform a subtraction at that point instead.

The way that we use the formula c squared equals a squared plus b squared may differ depending on which length of the triangle you're finding as well.

For example, here we have the same two questions again.

Let's see what it would look like if we use the formula each time.

With the left hand example, it would look like this, and with the right hand example, it would look like this.

Once again, these look pretty similar, but at what point do these solutions begin to differ? Pause the video while you think about it and then press play when you're ready to continue.

Well, these solutions begin to differ pretty early on, they differ here, and the difference is where we substitute the numbers that we have and where our unknown is.

When we are finding the hypotenuse, our unknown is c squared, but when we're finding on the shorter side, that our unknown is either a squared or b squared, and that will affect the way that we rearrange our equation.

So let's check what we've learned there, which calculations may be used as part of the solution for finding the missing length on this triangle? Pause the video while you make your choices and press play when you're ready for some answers.

The answer is c.

We would do 15 squared subtract seven squared somewhere while we are trying to find the missing length using Pythagoras' theorem.

How about this time? Very similar to last question, but the position of that 15 centimetres has changed now.

Which calculations may be used as part of the solution this time.

Pause the video while you make some choices and then press play when you're ready for an answer.

This time our answers are a and b.

We would either find ourselves adding together 15 squared and seven squared or seven squared and 15 squared when we're trying to find the hypotenuse on this triangle.

Those two calculations are equivalent.

It's over to you again now for task C.

This task contains one question, and here it is.

You have eight right engle triangles and you need to find the missing lengths.

But this time some of the missing lengths are a hypotenuse and some of the missing legths are the shorter sides of the triangle.

For any decimal answers you get, give them to one decimal place.

Pause the video while you work these out and press play when you're ready for some answers.

Let's see how we got on.

In question a, the missing length is the hypotenuse and it's 15 centimetres.

In part b, the missing lamp is a shorter side and it's 7.

9 centimetres.

In part c where looking for a shorter side which is 4.

5 centimetres.

In part d we're looking for the hypotenuse which is 3.

2 metres.

In Part e, we're looking for one of the shorter sides which is 35 millimetres.

In Part f we're looking for one of the shorter sides which is 40 millimetres.

In part g we're looking for the hypotenuse which is 2.

1 metres.

And in part h we're looking for two missing lengths.

Now, those shorter sides, they are equal to each other in length.

The way we know this is that the angles in that triangle are a right angle, 45 degrees.

The missing angle must also be 45 degrees, which means this is an isosceles triangle, so those two shorter lengths must be equal.

Therefore, once we use Pythagoras to work them out, we get not 0.

7 metres.

You're doing great so far.

We're now gonna apply what we've learned to use Pythagoras' theorem on a coordinate grid.

However, what makes this tricky is, sometimes the right angle triangles are not easy to see straight away.

Let's see how we get on.

Pythagoras' theorem can be used to calculate distances on a grid.

For example, if we want to find the distance between points A and B on this grid, how might we go about doing it? Sofia says, "I can't see a right angle triangle, so does that mean Pythagoras' theorem isn't useful here?" What do you think? Pause the video while you think about what Sofia is saying and consider how you might find the distance between A and B and then press play when you're ready to continue.

Well, Andeep says, "We can use the horizontal and vertical lines on the grid to construct the right angle triangle." We want to get this distance from A to B, and then using the vertical horizontal lines, we can create a right angle triangle either here or here.

Sofia says the length of the hypotenuse is a distance between A and B, and Andeep says, "We can use the squares to obtain the lengths of the horizontal and vertical edges." That is three units and four units.

Therefore, we can use Pythagoras' theorem like so to find the distance between A and B as being five units.

So if we can find the distance between two points and a grid, Pythagoras' theorem can also be used to find the distance between pairs of coordinates.

For example, how far is the four three from the origin? The origin as to coordinates zero zero.

Sofia says, "This question has no diagram at all.

Does this mean that Pythagoras' theorem isn't useful here?" What do you think? Pause the video while you think about what Sofia said and consider how we might find the distance between the four three and the origin, and then press play when you're ready to continue.

So this question is a little bit tricky because we have no diagram to visualise the right angle triangle.

But Andeep says, "We could draw a set of axes and mark roughly where four three would be in relation to the origin." Be about here-ish.

And then Sofia says, "If we draw a line from the origin to four three, a right angle triangle becomes clear to see." Let's do that.

We can now see again two right angle triangles.

Let's use this one.

Andeep says, "This is the same triangle that we used to solve the previous problem." The length of the horizontal side is four units, and the length of the vertical side is three units, and we know it's a right angle triangle because the x and y axes are perpendicular to each other.

So we can use Pythagoras' theorem again to find the length of the hypotenuse on this right angle triangle, which is a distance from point four three to the origin, and it would look a bit like this.

Once again, it's five units.

So what else can we do with Pythagoras' theoremm? Pythagoras' theorem can be used to find the radius of a circle on a coordinate grid when the coordinates of the centre and a pointer circumference are known.

Let's take a look, an example.

If we want to find the length of the radius here, how could we go about doing it? Sofia says, "We normally use Pythagoras' theorem when there are triangles not circles." So how can we use Pythagoras' theorem here? Pause the video while you think about this and press play when you're ready to continue.

Well, Andeep says, "The centre is at the origin." That's the zero zero.

"And the circumference goes through the point four three.

So the radius is a distance between these two points." That's this distance here.

Sofia says, "Once we draw on the horizontal and vertical distances, we have the same triangle as we had in the previous problems." Like so.

And then once we've done that, we can work out the radius of this circle by finding the distance between the origin and the point four three using Pythagoras' theorem like this.

Once again, it's five units, so let's check what we've learned.

Which triangle can be used to find the distance between a point with the coordinates four nine and the origin? Is it a, b, or c? Pause the video while you make a choice and press play when you're ready for an answer.

The answer is c.

That's because the distance between the origin and the point four nine is four in the horizontal direction and it's nine in the vertical direction, and that's what we've got labelled on this triangle here in the horizontal length and the vertical length.

Which calculations may be used as part of the solution for finding the radius on this circle.

Your choices are a, b, c, and d.

Pause the video while you meet your choice and press play when you're ready for an answer.

The answer is a or b.

Can help us find the distance between the centre and circumference, which is the radius on this circle.

Okay, it's O2 now for the final time for task D.

This task contains three questions and here they all are.

In each of these three questions, it's about using Pythagoras' theorem to find a distance on a coordinate grid.

Pause the video while you work through these and press play when you're ready to go through some answers.

Let's see how we got on with that.

Point P has coordinates seven 24.

How far is point P from the origin? It is 25 units from the origin.

Question two, what is the length of the radius of the circle? Well, the length is one unit.

And question three, the diagram shows quarter of a circle on a coordinate grid.

What is the value of a? We can see that a is the x value in the coordinates at that point, and that is 3.

2.

Now let's summarise what we've learned during this lesson.

The sum of the squares of the two short sides of a right angle triangle equals the square of the longer side.

That's what Pythagoras' theorem tells us.

The longest side of a right angle triangle is always opposite the right angle, and this is called the hypotenuse.

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

For example, we saw cases of that when we were working on the coordinate grid.

We didn't necessarily see the angle triangle straight away, but by drawing it on the grid or maybe even drawing a bit of a sketch of what the coordinates might look like, we could see the right angle triangle more clearly.

Thank you very much.

Have a great day.