a is half the size of the hypotenuse when 𝑥 = ?

The case of 30 and 60 degrees

Key Stage 3

Maths

Surds and trigonometry

What is the name of the longest side in a triangle?

Triangle Y has been enlarged. Work out the length of the base of triangle X.

﻿Hi there and welcome to another lesson with me Dr. Saada. In today's lesson, we will be looking at the very special cases of 30 and 60 degrees in right-angled triangles. All you need for today's lesson is a pen, a paper, a ruler and a protractor. So please pause the video and go and grab these, and when you're ready, come back and we'll make a start. I would like you to try this. Can you construct a right-angled triangle where a is half the size of the hypotenuse? Can you construct another one where b is half the hypotenuse? What is the size of angle x in each case? I'm going to give you a little hint here. Think about the explore task from lesson five, where we constructed similar triangles. If you need to, you can refer back to your notes from that lesson. Please pause the video to complete the try this task. Resume once you've finished. How did you go on with this task? Go on, say that to the screen. Really good. Okay, so there are so many possible solutions for this question. I'm going to show you the triangles that I managed to draw here. So I managed to draw this one here where I had the hypotenuse as eight centimetres and I had a as four centimetres, so I met that criteria. I measured the angle and the angle was 60 degrees. So now I know that a is half the size of the hypotenuse when x is equal to 60 degrees. And then for the second triangle what did you get as your x? Really good, because I drew the second triangle and I managed to get two centimetres for that side that was marked b, which is half of the hypotenuse four, when x was 30 degrees. So b is half the size of the hypotenuse when x is equal to 30. And this is really what we are going to look at in today's lesson. Those two specialist cases, when we have 30 degree angles and when we have 60 degree angles, what happens to the side length to the length of the sides and what happens in particular to the ratio between the sides in a right-angled triangle. If you look at this diagram, if you look at this diagram here, we have a right-angled triangle with one of the angles marked being 60 degrees. I want you to look at the hypotenuse where it says equal nine and the adjacent is equal to 4.5. These are the lengths of the two sides. Okay, what happens if I enlarge this triangle? So I'm going to keep that angle 60 degrees, but I'm going to enlarge the triangle and see, do I maintain that ratio between the adjacent and the hypotenuse? So let's have a look, I made it a bit bigger. Now the hypotenuse is 13.8 and the adjacent is 6.9. So that relationship is still there. The hypotenuse is still double the adjacent. Make it even bigger, I have the hypotenuse being 18 and the adjacent is still nine. So provided I have a right-angled triangle and that the marked angle is 60 degrees, then the hypotenuse, the largest side of the triangle and the adjacent side, so the side that isn't adjacent to the marked angle, they have that relationship or one of them, the hypotenuse is double the adjacent. It doesn't matter how big I make it. So I'm just trying to prove that to you here. There we go with one more, 24 is the hypotenuse, therefore the adjacent is 12. So now I should be able to tell you if I have a right-angled triangle and the marked angle is 60 and if I tell you that the adjacent is for example, 13, what would the hypotenuse be? You should be able to tell me that it will be double that, so that will be 26, okay? And this works whether I'm making my triangle bigger or smaller. So have a look at this one here, it's quite small. I have the hypotenuse 6.6, the adjacent becomes 3.3. So this is always true, if the malked angle is 60 degree in a right-angled triangle. If you look at this diagram here, we have a right-angled triangle with... And now let's look at this diagram here. This time I have a right-angled triangle, but the marked angle is 30 degrees, okay? So the marked angle is 30 degrees. And I'm going to look at the hypotenuse at the moment, it's 12 and the opposite is six. So this time the opposite is half the hypotenuse, okay? And I want to see does that relationship stay remains the same if I enlarge the triangle. So I'm going to make the triangle a little bit bigger, keeping that angle of 30 and observe what's happening to the length of the opposite on the hypotenuse. So the opposite is nine, the hypotenuse is still double it. So, make it a bit bigger. The hypotenuse is 21, the opposite is 10.5, it's half of it. And if I enlarge it even more, I have 12.94, the opposite double that is 25.8 and I can see here that the hypotenuse is 25.8. So, if we have a right-angled triangle and the marked angle is 30 degrees, the hypotenuse is going to always be double the side that is opposite to that angle, so the opposite side. So the opposite will be half of the hypotenuse or the hypotenuse will be double the opposite. So now, if I tell you that I have a right-angled triangle, and I tell you that, the marked angle is 30 degrees, and I tell you that the hypotenuse is 30. You should be able to tell me that the opposite will be the side opposite the angle will be? Excellent, will be half of that, so it will be 15. Okay, similarly, if I tell you what the opposite will be, you can tell me what the hypotenuse is by doubling it. Okay, so moving on from that demonstration and actually applying the knowledge from that into right-angled triangle is what we are going to do now. Just a reminder we're looking at right-angled triangle, the longest side is the hypotenuse, the side that is opposite the angle is a and the side adjacent to the angle, so where that angle is and the right angle on the same line that is a. so if we look at the first triangle, it's already been done for us, just to show you that we have a here, a is four centimetres, it's half the hypotenuse, okay? So it's half of that eight that we have there. Let's look at the second triangle. So we are given the hypotenuse is 16, we're given that the angle is 60. So what should that missing side be? Really good. So this side should be half of the hypotenuse and it should be eight centimetres. Moving on next one, we have been given a is 12.3, We're given that the angle is 60 degrees, and we want to calculate the hypotenuse and the hypotenuse is going to be? Excellent, double a. So the hypotenuse here is going to be double 12.3, which is? Really good, 24.6 centimetres. Now, next triangle, what did I do here? I just took it and rotated it. Is it going to make a difference? Excellent, it's not going to make a difference. And this is where you have to be really careful with identifying where is a and where is b, okay? So they just thought, this is the hypotenuse it's the easiest, it's the longest side. And this here is a, because it's the side that is adjacent to the angle, the angle that is marked for us, the 30 degree angle is there, that's the side next to it, so it's called a and obviously the other side is going to be the opposite. And it's opposite that 30 degrees angle, and it is 13 centimetres. So now we're given the 30 degree angle, so which two sides are we going to look at? Excellent. We're going to look at b being half of the hypotenuse. So if b is 13 and then hypotenuse must be 26 centimetres. Okay, next one. We are given a and this time, I wanted get to challenge you. So I thought, why not put some algebra in there. So we've got a, is equal to r centimetre. This is b, we're not being asked to calculate b. And this is the hypotenuse here and it is double r, so it's two times r and that is 2r because we have a 60 degree angle. Next triangle what do we have? We have a 30 degree angle, excellent. We're given the hypotenuse and we need to find the opposite. What's that relationship? This is the hypotenuse. This is a and this is b and I know that this must be half of the hypotenuse because we have a 30 degree angle accident. So the opposite is half of the hypotenuse. And last one, a bit tricky. We have a 30 degree angle. We're not given the hypotenuse. We're not even being asked to calculate the hypotenuse. This is really interesting because so far with the 60 and the 30 degree angle, we always had something to do with the hypotenuse. It's either the hypotenuse and a, or hypotenuse and b. Whereas this one, we've got nothing to do with the hypotenuse. We don't have to calculate it and it's not given to us. We are given this side here, which is eight adjacent. And we are given obviously b, we need to calculate it. We don't know what the hypotenuse is. Now, I can use my knowledge of the 30 degree angle to find the hypotenuse, right? So I know that the hypotenuse is going to be what? Really good, it's going to be 10 centimetres, it's going to be double the adjacent side. Now, how can I find b? Excellent. This is where we go and think about what are the other skills that we've learned before and we can apply them. And we've learned about Pythagaras. So I can use Pythagaras now to help me calculate that length of side b, okay? And we know what Pythagoras is. It's going to be x squared plus b squared, equals to 10 squared. And the best part of today's lesson is the independent task, where you actually get to practise what we have been talking about. So you have two questions and I would like you to read the questions carefully, understand them and answer them. For question one you need to calculate the missing sides and angles in the four triangles. If you want to challenge yourself, you can try and find that length of all the remaining sides and the angles. You will need between six to eight minutes to complete the independent task. So please pause the video now, and resume once you've finished. Off you go, Welcome back. How did you find that independent task? The answers to question number one are here for you so you can mark your work. I'm going to go through question two. Is it true or false, angle ABC is 60 degrees. Explain your answer. So first I'm going to start by labelling my triangle. This idea here is the hypotenuse. This is side a. I know that because I'm looking at angle B as my main angle or reference angle. So side AB is the adjacent and this side here AC is b. Now I'm going to look at the two sides here, and I know that the hypotenuse is double b, so angle ABC must be 30 degrees, not 60 degrees, okay? So it's false, the angle should have been 30 degrees, or I can say that AB should have been equal to 2.1, not AC. Well done if you had this correct. And for our explore task, I want you to look at the diagram given here, how many similar right angle triangles are there in this diagram? What relationships can you find between the side lengths within and between the triangles? And find as many missing sides as you can. I'd like you to pause the video and have a go at the explore task. Once you're ready, please come back and resume so we can mark the work together. Welcome back. How did you get on with that explore task? There are so many ways of answering this question. I'm just going to show you my thoughts on this task. So I looked at this and I started by looking at this small triangle and I could decide a and I thought when I know that one of the sides is one centimetre, I know one of the shorter sides, so I can use Pythagoras to help me find a, and I found a, the shorter side to be a square root of three out of two. I left it in surd form, we've recently done surds, and I know that it's more accurate than having to round it. The next step I did by looking at those two sides here, one of the sides is 0.5, the other it's a similar triangle, it's seven centimetres circled What have I done to get from 0.5 to seven, I multiplied by 14, so I enlarged it by scale factor of 14. So I need to do that to the other sides. So that one here, I have to enlarge it by 14 and becomes 14 centimetres, which tells me that this side here is 13 centimetres. I could do the same to the base of the triangle. So I had a, which is squareroot of three out of two. So I want to enlarge it by a scale factor of 14. So this whole thing now becomes seven root three. And if I want to find the small piece only without the a I can subtract, and that will give me 30 root three out of two. And you notice again, I'm leaving all my answers in surd form. To find the next part, I looked at 0.5, seven and 12, what's happening here? Well from 0.5 to 12, I multiplied I have a scale factor of 24. And now with knowing that the ratio from 0.5 to 12 we've multiplied by 24, I wonder how many other things you managed to calculate? Did you manage to find out the length of this side here or this side here, or did you find the length of whole base of the big triangle, then you calculate any angles. There are so many things that you could have done with this task? This brings us to the end of today's lesson. I would like you to have a little think about the things that we covered in today's lesson. What are the three most important things that you have learned, write them down, and please do not forget to complete the exit quiz to show what you know. Enjoy the rest of your learning for today and I will see you next lesson. Bye.

The case of 30 and 60 degrees

The case of 30 and 60 degrees

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UnitMaths / Surds and trigonometry

Maths