Varying the ratio of side lengths in right angle triangles

Key Stage 3

Maths

Surds and trigonometry

Which of the following is NOT equivalent to 2 × √9

﻿Hi there, and welcome to another lesson, with me, Dr Saada. In today's lesson, we will be looking at varying the ratio of side lengths in right angle triangles. All you need for today's lesson is a pen, paper, a protractor and a ruler. So please, pause the video now, go grab these, and when you're ready, come back and press play. Did you grab everything that you need? Brilliant, let's make a start. The try this for today's lesson is the following. Imagine a rod rotating about a point on a horizontal line. So you've got the background there, imagine that the green line there is the rod and that is rotating. How does the relationship between A and B change as you vary the angle X? What values of X would mean that A is longer than B, B is longer than A, A and B are the same length. If you're feeling super confident about this, please pause the video now and have a go at this. If not, don't worry, I'll give you some support. I'll be giving support in three, two, and one. Okay, so for support I want you to look at these diagrams here. So you have the original diagram there, the original location of the rod, and it's at angle X. Now I want you to think, if I start moving that, so let's move it. What happens? Okay, we're changing obviously the angle. So either making the angle bigger or we're making it smaller. So let's have a look at the second diagram. Imagine I take that green line, and I move it down a little bit. I'm making that angle smaller. Down a bit more. And a bit more. And a bit more. What's happening to A, and what's happening to B? Okay, think carefully. Really good. So, as we are making that angle smaller, A is becoming longer and B is becoming shorter. What about if I make the angle bigger, what will happen? So if I take, instead of making the angle smaller, if I start making it bigger, what happens, okay? So let's have a look. So imagine I take that line and I keep the same line, but I have made the angle a lot bigger. Really good, so larger angle is giving us a shorter A, and a longer B, because that line B will have to now move a bit forward, okay? In order for me to get that triangle there. So now with this hint, you should be able to make a start. Please, pause a video to have a go at the try this. Once you've finished, resume, and then we can mark and correct the work together. Welcome back. How did you get on with this task? Really good! So tell me, what did you get for the first part? What values of X would mean A is longer than B? What did you write down? Come on, say it to the screen. Really good! X is less than 45. If the angle is smaller than 45 degrees, A will be longer than B. What about B longer than A, what did you write down? Good job! So when X is greater than 45 degrees, B will be longer than A, and when are they both going to be the same length, A and B? Thinking about triangles and their properties? Really good! If it's exactly 45 degrees, because this reminds us of an isosceles triangle. If we want the two sides to be equal, then it must be an isosceles triangle. And the angles therefore, if it's right angle triangle, the two base angles should be 45 degrees. Really good job. Let's move on to our connect task. Let's have a look at the following triangles. What type of triangles are these? And how do you know? Just by looking at these, I've given you the side lengths of two of the sides, I've given you the length of the two shorter sides. You should be able to tell me that they are isosceles triangles because two sides are equal, excellent. So you will be able to tell me this. Also we know that in an isosceles triangle, also the base angles are equal. So if I take that protractor from the side here and I start measuring the angles, I should have 45 degrees angles, okay, apart from the 90 degree angle. So if I measure this angle here, it should be 45, and if I measure this one, it should be 45 degrees. This one in the second triangle, again 45 degrees, and 45 degrees. In the eight centimetre triangle, again, 45 degrees and 45 degrees. So I'm going to end up with 45 degrees in every single triangle, because interior angles in a triangle add up to 180, we subtract the 90, you're left with another 90, half it, because the two base angles are equal, you end up with 45 degrees. Okay. Now what other relationships are fixed by these angles? So we know, because we've got two equal angles, this fixes, obviously the side lengths. So each of them is the same. If one of them is four centimetre, the other is four. Also, I can say that the shorter sides, the two shorter sides have a ratio of one to one. So the ratio of the two sides are one to one. And this is what we are looking at in today's lesson, we're not just looking at measuring them, or the properties of the right angle triangles, but actually the ratio of the lengths in a right angle triangle. Okay, now I would like us to look at the following triangle. So I would like us to look at this triangle to start with. And I want to tell you that if A and B on this triangle are equal, so I start with, for example, five centimetre and five centimetre, then the angles will be 45 degrees each, okay? So this angle and this angle are 45 degrees each. I want you to think about this triangle, and I want to say to you, let's take it and let's imagine that I'm going to expand A. So I'm going to make A bigger. In fact, I'm going to double the size of A. So instead of five centimetres, I'm going to make A ten centimetres. So I am doubling the length of A, okay? I'm doubling it because I want to see this relationship, I want to check what happens if A and B, so these two sides, if one of them is double the other. So A is double the length of B. What happens to the angle? So if I get the protractor and measure the angle, now I have 30 degrees angle, okay? And if I think even about Pythagoras, and think about that longer side, I know that to find the hypotenuse here, I need to do five squared plus ten squared, and that gives me 125, then I need to square root it to find the length of this side here, so I know that this side here is the square root of 125. That's the length of it. Now, let's look at the second triangle, where A this time, so A here, is three times the size of B. So we're extending it even more, okay? So it's three times the size. If I measure the angle, the angle is even smaller than 30 degrees, it's 18 degrees this time. And again, I can use Pythagoras, to find the length of the hypotenuse, and if I do this, it's 15.9. Now let's look at the next triangle. Again, I have a right angle triangle, and I have this time made the length of B double A. So now B is longer, okay? It's ten, it's more than five, it's double five. So what happens to the angle in this case? So if I take a protractor and measure it, the angle this time is obviously bigger, it's going to be 60 degrees. Again I can do the same thing if I really want to find the side of, the length of the hypotenuse, and it's 11 centimetres. I'm just showing you everything that we can do with those triangles that we have now. Okay, and the idea is that in today's lesson, we're going to be looking at, if we change these angles, what happens to the sides? And in particular, what happens to the ratio of one side to the other? So the ratio of one side to the other, you divide one side by the other, and this is what I want you to do in the independent task. So for your independent task, I would like you to draw four right angle triangles, such that each triangle has a 30 degree angle. So it's a right angle triangle. So one of the angles is 90 degrees, the other is 30 degrees. Name them A, B, C and D. It's entirely up to you which one you name which. It really doesn't make a difference. It's entirely up to you how big or how small these triangles are. Just make sure that they are not the same length, and make sure that the sides are more than one, so they are sensible enough for us to use for the calculation. For each triangle, I want you to label the hypotenuse with M, label the opposite side to that 30 degree angle with N, and label the last side with O. So let me go through this on the diagram here, so you roughly know what I mean. For each triangle, label the hypotenuse. The hypotenuse is the longest side. So look at the longest side, which is opposite the right angle, and label it with M. You can see here on mine, I labelled mine with M. Label the side that is opposite to the 30 degree, so the one that is opposite to the 30 degree is N. So this is 30 degrees, so this one, I labelled it N. So that's how you label yours. And the last side, obviously just label it with O. Then I want you to measure the sides and complete the table. Okay, so in the table you have Triangle A, B, C and D. I want you to write down the marked angle, so each of the angles is going to be 30, because I'm asking you to have a 30 degree angle in each. I want you to measure the sides M, N and O, and list them down. Then I want you to find the ratio of N to M, and you do that by dividing the length of N divide by the length of M. Then I want you to find the ratio of N to O, so divide the length of N by the length of O. We should start to see some pattern there. When you're done with this, I would like you to enlarge one of your triangles, by a scale factor of two, and by a scale factor of a third, to notice what happens there. So now, it's time for you to get on with the independent task and complete it. It should take you about 15 minutes to complete. So please pause the video to complete the task, resume once you are finished. Welcome back! How did you get on with it? Okay, really good. So, you will have four different triangles to mine, it's going to be very, very unlikely that we ended up with the same triangles. These are the ones I drew. I wonder which ones you did? Did you make yours a bit smaller, a bit bigger? Okay so I completed the table like this, so I had, for my first one, a 30 degree angle, and the three sides were 4.9, 9.3 and 8.1. The ratio, so when I divided N by M, was 0.5, rounded to one decimal place. And the ratio of N to O, so when I divided N by O, the ratio was 0.6. I repeated the same to Triangle B, and I had slightly different sides. It was a bit bigger, the triangle was a little bit bigger, but again, with the ratio, I ended up with 0.5 to one decimal place for the ratio of N to M. And then the ratio of N to O, so the ratio of the two shorter sides to one another was again 0.6. Then I did the same for C, and I decided to go for a much smaller triangle this time, so the sides were a bit smaller. The ratio still, to one decimal place, was 0.5, and the ratio of N to O was still 0.6. Again, it was not exactly 0.6, that's me rounding to nearest one decimal place. And for D, I went for an even smaller triangle, and I still ended up with 0.5 as the ratio of N to M, and the ratio of N to O was 0.6. Okay? So, I wonder if you ended up with similar ratios to mine. The second part required you to enlarge one of your triangles by a scale factor of two. So taking that triangle, and doubling the sides. You will maintain the same angles, but you will make the sides twice as big. Again, when you do the ratio for that, it should have given you exactly the same ratio as the original triangle that you enlarged, it shouldn't have been different. Scale factor of a third means you're making it a bit smaller, so you're making it three times smaller, and again the ratio of the sides should have just stayed the same. Now, what are these ratios really telling us? So if we look at this here, the ratio of N to M, okay, so the ratio of N to M, N to M is always the ratio of one of the shorter sides to the longest side. Which particular side? It's always the side that is opposite to the 30 degree angle that you drew. So in this one here, again, ratio of N to M was N divided by M, where N was again opposite the 30 degree angle, divided by M. So what we see from this task is that if I have a right angle triangle, and I have an angle of 30 degrees, the side that is opposite to the 30 degree angle is always half the size of the hypotenuse, it's always half of the length of it. So every time I have a 30 degree angle, right angle triangle, the hypotenuse is double the opposite side. Or the opposite side is half the hypotenuse. We are going to continue to explore right angle triangles, and see what happens when we vary the side lengths. And in order to do this, I want you to construct a right-angled triangle, similar to the one here on the screen, such that X is equal to 30 degrees. So I want you to construct a right angle triangle, so one 90 degree angle, the other angle has to be 30 degrees, and I want the hypotenuse to be four centimetres. If you're not too sure how to construct triangles, can you please go back and watch the video for constructing triangles, and then come back here. Now once you've done the first triangle, I want you to construct another one, where again, we still have the four centimetre for the hypotenuse, but I want the angle to be 45 degrees, and now I want another one, so a third one, where the angle is 60 degrees. So we have 90 degrees, 60 degree and a four centimetre hypotenuse. And I want you to make some observations about A, B and four. Okay so when you are ready to make a start on this explore task for me, please pause the video and complete it. Resume the video once you are finished. This task should take you about seven minutes. Off you go. Welcome back! How did you find this task? It's a really interesting one, isn't it? And it's a good opportunity for you to recap how to construct a triangle, so really good. Let me show you what I did. So I started by drawing the first one, I had 30 degree angle, and I had four centimetres as the hypotenuse, and obviously a right angle triangle. And this is what I noticed, that this side here was half of the hypotenuse. It was two centimetres and the hypotenuse was four. So I can say that B is half four at 30 degrees. So B was half of the hypotenuse at 30 degrees. The second one I drew was this one. So I had four centimetre, and the other two sides, when I measured them, I had 2.8 and 2.9. And I had 45 degree angle, so I can say at 45 degree angle, A and B are almost equal. So I have 2.8 and 2.9, they're very, very close to one another. And the next one, this is what I did. So I had four centimetres as the hypotenuse, and what I noticed is, this time, this side here was half of the hypotenuse. So, I noticed that at 60 degree, A is half four. So at 60 degrees, the adjacent side to the angle, to the angle 60, was half of the hypotenuse. Did you get something similar to me? And did you make similar observations? Really good, well done. So if I want to summarise, what I found out from this explore task, for me, I found out that if I have a right angle triangle and I have a 45 degree angle, then the two shorter sides are equal. I found out that if I have a right angle triangle, and I have an angle of 30, that side opposite to that 30 degree angle is half of the hypotenuse. I found out that if I have a right angle triangle, and a 60 degree angle, then the hypotenuse is double the side that is adjacent to the angle. So if the angle is here and the right angle is here, the side that is connecting the right angle and the 60 degree angle is half of the hypotenuse. And that's a really important observation for me. I also learned from today's lesson that it doesn't matter how big I make the triangle, as if I maintain those angles, the ratio of the sides stays the same. And in next lesson, we're going to take this idea about the ratios of the triangles, when we have 30 and 60 degree angles, and explore it a bit more. This brings us to the end of today's lesson. A huge well done for your resilience and effort during this lesson. Please show what you know by completing the exit quiz. This is it from me for today, I will see you next lesson. Bye!

Varying the ratio of side lengths in right angle triangles

Varying the ratio of side lengths in right angle triangles

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UnitMaths / Surds and trigonometry

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