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

﻿Hello and welcome to this lesson, with me Dr. Saada. For today's lesson, we will be looking at Generalising Pythagoras's Theorem. All you need is a pen and a paper. Please take a moment to clear away any distractions and try to find a quiet place where you will not be disturbed. Okay. So when you're ready, let's begin. So to start off with, I would like you to have a go at this question, how many numbers one to 100 can be expressed as the sum of two square numbers? you can use the same number twice. For example, two squared plus four squared is equal to 20, two squared is a square number, four squared is a square number and the sum of that 20 is an answer or a number between one to a 100. Another example would be, three squared plus one squared equal 10. Pause the video and see if you can have a go at this. So pause the video for me in three, two, one. There are lots and lots of numbers that you could have used. I used three squared plus two squared equal 13. I used five squared plus two squared equals 29 and seven squared plus six squared equal 85. There are lots of numbers that we can use to make sums from the square here. It's really important to notice that our numbers do not add up to more than 100 because that was the constraint that I gave you. For example, five squared plus nine squared equals 106 and that is greater than 100 Now let's have a look at this question, find the area of each square and complete the table. In the diagram here, you have a right-angled triangle surrounded by three squares, a blue, a pink and a green one in each case. If we look at the first one, and we try and work out the area of the squares, the green square is a three by three, which gives us an area of nine units squared. The pink one is four by four, and that gives us 16 units squared and the blue one we've done this in lesson three and lesson four, when we looked at tilted squares, I can here, divide into four smaller triangles each of them with an area of six plus the one square in the middle, that gives me a total area of 25 units squared. If I go to the next one, the green square is a two by two, it gives me an area of four units squared. The pink squared is a three by three, which gives me nine units squared and if I do the same thing with the blue one, it gives me an area of 13 units squared. Moving onto the last one here, the smallest one, If I look at the green square, I divide it into, I can divide it into four small triangles and if I join each two triangles together, they gave me one unit squared, which gives me a total area of this square being two, for the pink one it's exactly the same, so it's an area of two units squared and for the blue one, I have a two by two, which gives me four units squared. What do you notice about the area of the blue square? If you look carefully at the table here, what do we notice? If I add nine plus the 16 that gives me 25, four add the nine gives me 13 and two add two gives me four. So I notice that the area of the blue square is equal to the sum of the area of the green and the pink squares together. And now I can make my language a bit more precise and I can say, that I notice that the sum of the squares of the shorter sides, so the square here for the shorter side, let me get my pointer. So you've got the shorter side of the triangle here, the square of the shorter side of the triangle and the other shorter side. So the sum of the squares of the shorter sides of the triangle are equal to the square of the longer one and this is our longer side of the triangle. This one here. You may be wondering, why do we need this? Or what if I give you a triangle and I ask you to find the length of one of its sides? Knowing the others or the squares of the other sides will help us, and that's what we will be looking at in today's lesson. And if we look at this diagram here, it's very similar to the one we've seen before, the only difference is I'm using A, B and C to make some generalisation. What we discussed from the previous task is that the square of the shorter side, so this is my shorter side, this is the shorter side of the triangle, so these are the two shorter sides of the triangle. We discussed that the square of this one, plus the square of this one, it's going to be equal to the square of the longer side. We're going to look at the language that we're using and just improve it a little bit now. We are going to say that, in a right-angled triangle, the sum of the squares of the shorter sides is equal to the square of the hypotenuse. What is the hypotenuse? The hypotenuse is the longest side in a right-angled triangle. It is the side opposite the largest angle in a triangle, In this case the hypotenuse is opposite the right angle, 'cause the right angle here is the biggest angle that we have in this triangle. The other two angles are acute angles. Now instead of saying those really long sentences, how can I make generalisation using algebra for this? I can say if this side here is A, then the area of this square is A times A, which is A squared and if the side here is B the area of this square is B multiplied by B, which is B squared. Similarly for this square here, if the side length is C, C multiplied by C is C squared. So I can say that A squared plus B squared is equal to C squared. What other equations can we write? Pause the video and have a little think Okay. If I know that A squared plus B squared is equal to C squared, I can also say that B squared plus A squared is equal to C squared. I can also say that B squared is equal to C squared minus A squared. And I can also say that C squared minus B squared is A squared, equals A squared. Okay. And now let's have a look at this with some numbers. So if I tell you that the area of the bigger square here is 50 units squared. Can you tell me what could the area of the two smaller squares be? This one could be 20 and this one could be 30. I could have gone for 25 and 25 if those two sides were equal. I could have gone for 15 and 35. It doesn't really matter as long as the area of this square, plus the area of this square equal the area of the bigger square. How is that going to help us find out the length of the sides of the triangle? let's have a look at this. If I know that the area of this squared here is 20, can I work out the length of this side? It must be square root of 20 the square root of 20 multiply by square root of 20 is 20. If this area here is 30, can I find out what number multiplies by itself to give me 30? Then B must be square root of 30. And you've done this in lesson two. And now I can also work on this side here. If the area of this square is 50, then the length of the side C must be square root of 50. And now it is time for you to independently practise some of the skills that we have looked at. I would like you to have a go at questions one to three. So pause the video and have a go at this. Okay, well done for having a go at these questions on your own. Now let's have a look at the solutions. Please have a pen handy, so you can mark your work and correct it as we go along. Mark the hypotenuse on each of the following right-angled triangles. Now we said that the hypotenuse is the longest side of the triangle and it is opposite the right angle. So if I look here, this is the right angle. One arrow there drawn for you to show you where the hypotenuse is. So the hypotenuse is M N here. Let's look at the second one, the right angle is here. The side opposite to it, is Q R. So Q R is the hypotenuse. If you had this correct well done, if not, please correct it. Question number two, Fill in the gaps on the diagrams below. And we've done a question similar to this earlier on during the video. The area here is 80 units squared. What could we have as the areas of the two smaller squares? I could have 35 and 45. That gives me a total of 80. If I look at the second one, I gave you one of the smaller ones I left the other two for you to do. And I chose 36 for this one. I did the math to give me 48. I wonder what numbers you have used as there are lots of possible answers for this question. And moving on now, question three, find the length of the missing side of each triangle. I'm going to go through this question with you step by step. Okay. And now let's have a look at question number three. I absolutely love questions like this. I have a right-angled triangle. So the first thing I'm going to say to myself is which one is the hypotenuse? This is my right angle. So this side here, that I need to work out is the hypotenuse. And I've been given the two shorter sides. What have we learned today? That the square of the two shorter sides So the square of this side, four squared plus the square of this side, three squared will give me the square of this side. So let's write that down. We started with A squared plus B squared is equal to C squared. Really doesn't matter which side here of the two shorter sides I use as A and B. All I know is the sum of the two shorter squares. So instead of A squared, I'm going to write three squared for this short side, plus the four squared. So the square of this side is equal to the square of this side. And I don't know what this side is. I'm going to work it out. What is three squared? Three squared is nine plus four squared is 16. Altogether is equal to C squared. What is nine plus 16? That gives me 25 is equal to C squared. Remember, this is really important. It's equal to C squared, I don't know the length of this side now. I only know that it's C squared. What number multiplies by itself to give me 25? That is the same as saying, what is the square root of 25, which is five centimetres. So C here is five centimetres. Okay. And now let's have a look at the second triangle, which is a little bit more challenging. We have been given two sides again, the length of two sides, one of them is five centimetres One is two and one is a question mark. We need to work that one out. So I will always start as usual with identifying where's the hypotenuse of this triangle. So this is my right angle. So the side opposite to it is five. And that tells me that this is the longer side of the triangle. It's the hypotenuse. I know that the sum of the squares of the two shorter sides. So two squared plus something squared is equal to five squared. So I can start writing this down. I would write A squared plus B squared is equal to C squared. I don't know what A is, I don't know what this side is so I'm going to keep it as A squared. I know that the other shorter side is two. So the A squared of that is two squared. So A squared plus two squared is equal to five squared. Now, if I know what this side I know five squared, and I know the two squared, I can work this out by changing, by re-arranging my equation, or by just saying that A squared is equal to five squared, subtract the two squared. Now five squared is 25, subtract four that gives me A squared equal to 21. So I know that the square here, if I was going to draw one, the square here will have an area of 21. What would be the side length? Well I need the number that multiplies by itself to give me 21. And therefore it is the square root of 21. And it's really important to know that the side length of a triangle does not always have to be a whole number. In this case, we're leaving our answer in form. So we're leaving it as the square root of 21. You can plug that into the calculator and write it down as a decimal, if the question asks you to do so Okay, And now I would like you to have a go at this explore task. For each right-angled triangle write some equations relating the side lengths. Pause the video and have a go, unless you want a hint. I will be giving a hint in three, two, and one. Going to go through the first one, just to help you figure out what to do for the others. So if we look at this triangle here, I have a triangle that has side lengths, D E and F. I will always start by looking for my right-angled triangle right angle and the hypotenuse. So you've been told here that this is a right-angled triangle in the question so you know that my right angle is here. Let's just make this. And I know that this side here E is the hypotenuse of the triangle. I can write down that D squared plus F squared is equal to E squared. The sum of the square of the two shorter sides is equal to the square of the longer one. I can also say that D squared is equal to E squared minus F squared. And so on. My second hint is having a look at those two dashes there. What do these dashes mean? When do we use them on sides? Yes, we use them when the two sides are equal. So this tells me that this is an isosceles triangle. Okay. Pause the video. And with those two hints, you should be able to make a start now. And, here are some of my answers for the first one P squared plus Q squared equals R squared, Q squared plus P squared is equal to R squared. It really doesn't matter which way I add them. I can say that R squared is equal to P squared plus Q squared. I don't have to have the equal sign only at the end. I can write subtraction ones. So I can say that R squared minus P squared equals Q squared. And for the next one, I can say that N squared plus six squared equals N squared. So notice the difference here. Instead of having another letter for the lines of this short side, I decided to you was number six instead of that. And that's fine. I can also write that six squared is equal to N squared minus M squared. I can say that's six squared plus M squared is equal to N squared, and I can have another subtraction one where I say that M squared is equal to N squared minus six squared. I could have written so many more. I wonder which ones you have written in your books And looking at this last one here. It's a really interesting one. As I said earlier, these two dashes here mean that the two sides are equal. So this side, if this side here is K, then this side is equal to it. And therefore must also be K. I can say that K squared plus K squared equals J squared. I wonder if any of you wrote two K squared equals J squared 'cause that would also be correct. K squared is equal to J squared, the square of the longer side minus K squared. I could have also said that J squared minus K squared equal K squared. So a huge well done if you had these correct. Okay, that brings us to the end of today's lesson and a really big well done on all the fantastic learning that you've achieved today. I've got three things for you to do. Firstly, I would like you to look back at your notes from today and identify the three most important things you've learned today. It's entirely up to you what they are. Secondly, I would like you to have a go at the quiz. Finally, if you are able to, please take a picture of your work and ask your parent or carer to share that with your teacher so they can see all the fantastic things that you've learned today. And if you'd like, you can ask your parent or carer to send the picture of your work to @OakNational on Twitter, so I can see your lovely work too. Well, all that's left for me to say is a big thank you. And I'll see you in the next lesson. Bye.

What is the name of the longest side of a right-angled triangle?

Which diagram shows the hypotenuse correctly highlighted and labelled?

A right-angled triangle has the sides a, b and c. Choose the correct formula for finding the length of side a.

Work out the length of the hypotenuse of the triangle shown below.

Generalising Pythagoras's theorem

In this lesson, we will develop an understanding of Pythagoras' theorem by drawing upon our tilted squares knowledge.

Maths

Y9 Maths

Key Stage 3

Year 9

Generalising: Pythagoras's theorem

Pythagoras's theorem

Geometry

THEME

UNIT

Tilted squares

Surds from tilted squares

Finding the length of a line from tilted squares

Pythagorean triples

Pythagoras: Finding right-angled triangles

Pythagoras's theorem on the Cartesian plane

What is the area of the square shown below, in square unit?

The digram shows a right-angled triangle. Which of the statements is correct?

Use Pythagoras’ Theorem to determine which of the sets of numbers below are Pythagorean triples:

The right-angled triangle ABC has short sides of length 8 cm and 4 cm. Find the length of AC.

What do you need to do in order to find the area of the triangle shown below.

Find the distance between the points (2, 1) and (4, 3).

What's the length of one side of the square  shown below?

What is the length of the line below?

Lesson overview:Generalising: Pythagoras's theorem

Core Content

Pythagoras's theorem: