Look at the diagram below and fill in the gap: The area of the larger square is equal to the ...........  of the area of the two smaller squares.

What is the side length of the larger square?

If the pink triangle has a base of 10 and a height of 2. What is the area of the green tilted square?

If the pink triangle has a base of 10 and a height of 2. What is the side length of the green square?

Right-angled triangles and tilted squares

Key Stage 3

Maths

Pythagoras's theorem

The blue and red lines are equal in length.

The blue line is longer than the red line.

The blue line is shorter than the red line.

Which statement best describes the blue and red lines shown in the diagram?

What is the length of the missing side of the triangle, labelled n?

What is the length of the missing side of the triangle, labelled m?

﻿Hi there. And welcome to another Maths lesson with me, Dr. Saada. In today's lesson, we will look at right angled triangles, and tilted squares. For this lesson, you will need a pen and paper and a ruler. So if you do not have these handy, please pause the video, go grab these. And when you're ready, we can make a start. Your first task for today's lesson, is to find the area and side lengths, of these three squares. So in the first diagram, you've got a grid, and you have a tilted green square. In the second diagram, you have a grid and you have two squares. Two green squares. What do you notice about them? What's the same and what's different between the two diagrams? Please pause the video, and have a go at this if you're feeling super confident. If not, I'll be giving you a hand, in three in two, and in one. Okay, so as you remember what we have done in the last two lessons? We have been looking at areas of tilted squares. We looked at three different methods, in order to help you find the area of a tilted square. You can split the shape into smaller shapes, making triangles and squares. Calculate the area of each and add them up. You can count the squares if you want to. Or you can calculate the area of the grid, subtract the area of the four triangles, the white ones there. And, you have the area of the tilted square. Now, with this hunch, you should be able to make a start. Off you go. Please pause the video to complete the Try this task. This should take you about five to seven minutes to complete. Resume the video once you're finished. There are lots of things that you could have noticed with this task. I've uploaded some of these, these diagrams here for you. For example, the area of this square here, is 13. And the areas of the two smaller squares, are four and nine. The triangles here are congruent. So the pink triangle is congruent to the pink triangle, in the second diagram. The orange is congruent to the other orange triangle. The purple ones and the blue ones. And in fact, all the triangles are congruent to one another. Which means, all the side lengths are equal, and the angles are equal. The areas of the triangles are the same. The area of the square, the bigger square, is equal to the sum of the area of the two smaller squares. They area of the overall grid is the same. And now I just took those two sections, out of the first and the second grid. So from the first one, I took the pink triangle and the square, that is connected to it. And then from the second one, I took the same pink triangle and the two squares that were connected to it. Now, if we start by looking at this here, we have a square of four. The area of the square is four. Which tells me that this side length here, must be two. Because square root of four is two. And this square here, has an area of nine. And this tells me that the side length here, is going to be the square root of nine, which is three. Now, this square here has an area of 13. Which tells me that, this side length must be the square root of 13. And this is something that we looked at, in our previous lesson. Now, these two triangles are congruent. It's exactly the same triangle. So this tells me that this side here also, is two and this side here is three. Now, what you notice from this is that, the area of the largest square, is equal to the sum of the areas of the two smaller squares. So the 13 is equal to the four plus the nine. And how is that going to help us? Well, I know that this pink triangle is the same. So if I knew this side length here, and I knew this side length here, I should be able to have this side length from them. Because I should know that, if I draw a square around the two here, that should give me an area of four. And if I draw a square around this side here, it should give me an area of nine. The nine plus the four is 13. So that tells me the area of the larger square that is connected to the slanted side, which is 13. And therefore, I can use that to help me work out, that the side length, of the slanted side of the triangle, is square root of 13. This is what we're going to look at in today's lesson. So let's have a look at this in a bit more depth. Let's have a look at the two diagrams that we have here. We said that the area of the larger square, is equal to the sum of the area of the two smaller squares. So if I tell you that the side lengths here, in this one, for the pink triangle are one and four, you should be able to tell me, the area of this smaller square is one. Because one squared is one. The area of this square here is four times four, which is 16. Therefore the area of the tilted square and the first one, the bigger square should be one plus 16, which is 17. If I know the area of the bigger green square, I know that the side length of that is the square root of 17. So let's have a look at this, this example. I have been told that the length, of this side of the triangle is five, and the length of the other side is nine. I want to know the length of this side here. Now, I know that this pink triangle here is the same as this pink triangle here. They are congruent. Now, I also know that, the area of this square here, is equal to the sum of area of this one plus this one. So if I know the area of this square here, I can then use that to help me find the side length. Because I know all I have to do, is to square root it. If I find the side length here, it's the same as this side length here. So I will have found out, the length of that third line, or third side of the triangle. So let's make a start. The very small square here has an area of 25. The second one I have nine. So if I square it, that's 81. So this square here has an area of 81. Therefore the bigger square, has an area of 81 plus the 25, which is 106. Now that I know the area of the bigger square, I can use that to find the length of the missing side. This side is going to be the square root of 106. And because I know this side, and I know that these two pink triangles are congruent, now I know that this side here must be square root of 106 too. Now it's time for you to put all of that, into practise with this independent task. Let's read it together. In each pair of diagrams, all of the triangles are congruent. For each set, find the area of each square, of each green square. And find the length of the final side of the triangle. So you need to find the area, of this square here, to help you find out, the length of this side of the triangle. Now, in order for you to do this, you need to use the second diagram, and the area of the two smaller squares. You need to repeat that for each set here. Please pause the video to complete the independent task. This should take you roughly 10 minutes. Resume the video once you're finished. Welcome back. How did you get on with the dependent task? Really good. Let's mark and correct the work together. So, in the first diagram, we can see that we have a triangle, that has one side length of one centimetre. And another side length of three centimetres. We want to find the length of the longer side of the triangle, or the missing one on the triangle. So if I use the second grid, I can label the one and the three, the sides that have been given to me. Because these two triangles are congruent. Now I can work out the area, of the smaller squares. So this area is one and this gives me, an area of nine. Add them up. This gives me an area of 10. And I know that the area of the larger square, is equal to the sum of the area, of the two smaller ones. If this area is 10, then this tells me that, this side length is the square root of 10. Next one, we have a seven and 10 centimetres, have been given to us as to side lines of the triangle. Now, we use that congruent triangle in the second shape and find the area of the smaller squares. The area of this square is 49. The area of this square is 10 squared, which is 100. If we add them up, hundred plus 49 is 149. And therefore, the length of the missing side, or the length of the third side of the triangle, is 149. It is the same side for the square, basically, and the triangle. How did you get on with this? Did you get the answers correct? Very good. Next one. We had seven and six centimetres. So if we find the square of six, 36. The square of 7, 49. Add them up. This gives us the area of the larger square, as 85. To find one length of the square, that's the square root of 85. And because it's a square, all the sides are equal. So obviously, if this side here, is 85, then square root of 85. Then this side is also square root of 85. Next one. We had a three and nine, as the side lengths of the triangle. So, we use congruent triangle. Gives us the area here of nine and 81, for the smaller squares. Add them up. 81 plus nine is 90. And this tells me that, this side has a length of square root of 90. Did you get all of this correct? Huge well done. It's time now for our explore task. You have here two diagrams. If I tell you that the pink triangle has a base of 10 and the height of three, what can you work out in the diagrams here? Have a little think of all the things that you might be able to calculate. Okay? Right. So, this students here says,' I can work out the other side, length of the largest green square.' This student says,' I can work out the total area of the triangles.' 'I can work out the perimeter of the diagrams.' 'I can work out the area of the largest green square.' 'I can work out the area of the smallest green square.' 'I can work out the area of the pink triangle.' So there are lots and lots of things that you can do with these two diagrams. And the fact that the base of a big triangle is 10. And the height is three centimetres. Now I'd like you to have a go at answering all of these. Can you work out all of these? Can you find out more information? Off you go. Please pause the video and spend, between 10 to 15 minutes on the Explore task. Resume the video once you're finished. How did you get on the Explore task? Did you manage to work out all the things that the other students were saying that they can work out? Okay, let's have a look at this together. The first one, 'I can work out the perimeter of the diagrams.' Well, I know that this side here is 10, because I know one side of the pink triangle is 10. And this one here is three, because the other shorter side is three. Now, this means that this side length, the total is 13. If I want to find the perimeter of the diagram, then I need to go all the way around the shapes. I need 13 plus 13 plus 13 plus 13. Or 13 multiplied by four, which gives me 52 centimetres, as the perimeter of one of the diagrams there. Next one, 'I can work out the area of the pink triangle.' Well I know the base and I know the height. I can use the formula, based multiplied by the height divided by two. That's 10 multiplied by three divided by two. And that should give me 15 centimetres squared. 'I can work out the total area of the triangles.' Well, one pink triangle is 15 centimetres squared. Know that all of these triangles are congruent. So if I want to find the total area of the triangles, all I have to do is multiply 15 by four. And that will give me a total area, of 60 centimetres squared, for the area of the triangles. What else can we do? 'I can work out the area of the largest green square.' 'I can work out the area of the smallest green square, and the side length.' So let's have a look. I know that this side is three and this is 10. So, we can work the area of the small square, here being nine and here it's hundred. I know that the area of the larger square, equal to the sum of the area of the two smaller squares. If I add the nine plus 100, that gives me 109. That tells me the area of this square here. And if I know that the area of this square here is hundred and nine, I know now that, one of the side lengths must be square root of 109. So now I know that this side here, is equal to this one. It's same pink triangle. They are congruent. So there are lots and lots of things, that you could have calculated. Only by knowing that, the pink triangle had a base and a height of three and 10 centimetres. This brings us to the end of today's lesson. A huge well done on all of your efforts. Please remember to complete the exit quiz to show what you know. Enjoy the rest of your learning for the day and I'll see you next lesson. Bye

Right-angled triangles and tilted squares

Right-angled triangles and tilted squares

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UnitMaths / Pythagoras's theorem

Maths