Lesson video

Hi, my name's Miss Kidd-Rossiter.

And I'm going to be taking you through this lesson today on nested triangles.

This is going to build on the work that we've already done on similar triangles.

If possible, make sure you're in a nice quiet area and you're free from distractions so that you can concentrate on this lesson.

If you need to, pause the video now, so you can get yourself sorted, if not, let's get going.

So we're going to start today's lesson with a try this activity.

Your job here is to find all the missing angles in the triangle.

So you've got four triangles on your screen to find the missing angles.

And then, once you've done that, what do you notice? So pause the video now and have a go at this activity.

Well done, having a go at that task.

Let's have a look at it together now.

So, we know here that this angle ACB is 90 degrees.

Hopefully you then used a protractor to figure out that this angle here CAB was about 26 degrees.

And if you measure this angle here, angle ABC, you should have realised that that was about 64 degrees.

Remember if you got a couple of degrees, either side.

That's absolutely fine.

So now let's look at triangle PQR.

So again, we've got a 90 degree angle here.

Right angle.

Again, when we measured these, we should have realised that this angle here, was 26 degrees and this angle here was 64 degrees.

What do you notice about these two triangles? So we can say, that the corresponding angles in ABC and in PQR, are equal.

Can we notice anything else about these triangles? I notice here, that the height of triangle ABC, is two squares and the base of triangle ABC is four squares.

What about triangle PQR? Well, the height here, I can see is four squares and the base I can see is eight squares.

So triangle PQR is an enlargement of triangle ABC, scale factor two.

What does that mean about these triangles? Tell the screen.

Brilliant, they're similar.

So triangle ABC and triangle PQR are similar.

So that's just a recap on what we did last lesson.

Now let's have a look at this diagram here on the right hand side.

So triangle DEF and triangle DHJ.

So again, in each triangle, we've got a right angle.

We've got the angle here, which is 26 degrees.

And if you notice that angle, is common to both DEF and DHJ.

Is in both triangles.

Then in our yellow triangle, DHJ, we've got this angle here, which is 64 degrees and we've got the same angle here, 64 degrees in our blue triangle.

So again, we can see that we've got the same corresponding angles in the triangles.

So now let's see if one is an enlargement of the other.

So for HJ, the length of this side is 2.

5 squares, isn't it? What about the length of EF? What's that? Well, it looks like it's 1.

5 squares to me.

And then here DF is three squares and DJ is five squares.

We can say that for our blue triangle, the base to the height is in the ratio three to 1.

5 for our blue triangle.

For our yellow triangle, the base to the height is in the ratio five to 2.

5.

And we can see that these two ratios are equivalent.

They're both the ratio in its simplest form two to one.

So we know that the yellow triangle is an enlargement of the blue triangle, and therefore these triangles are similar.

So we're now going to apply this learning to the connect part of the lesson.

So you've got this question on your screen, explain why triangles, ABC and DCE are similar.

I'd like you to pause the video now and take in this question, read it, figure out anything that you can, look at the hint that's on the slide, and then when you're ready to go through it, resume the video.

Okay.

So, the first thing that I noticed was that I've got a pair of parallel lines here.

So AB, is parallel to DE.

How do I know that? Tell the screen.

Great.

The arrows on the lines indicate that the lines are parallel.

So now I'm going to look at the hint that was given to me.

What do I know about angle ABC and CDE? So angle ABC, is this one here and angle CDE is this one here.

What do we know about those? Good.

You should have realised that angle ABC and angle CDE, are alternate angles.

What do we know about alternate angles? Good.

You should've realised that alternate are equal.

So that means that we can mark these with a line to show that those two angles are equal.

Have I got another pair of alternate angles on this diagram? Yes, I have, you're right.

Angle CAB and angle CED are also alternate.

So I can write that as my sentence, angle BAC and angle CED are alternate angles.

And we know that they're equal.

So finally, what about angle ACB and DCE? These two angles here.

Why are they equal? Good.

They are vertically opposite angles.

So those are equal as well.

So angle ACB and angle DCE are vertically opposite.

So they're equal.

So that means that the corresponding angles in both triangles are equal.

So that means that one must be an enlargement of the other, and therefore they are similar.

So we're now going to have a go at working out the missing lengths.

I don't know about you, but I would find it helpful to redraw one of the triangles so that they're both in the same orientation.

I'm going to redraw triangle ABC.

So there's in the same orientation as triangle CDE.

So here it is.

I'm just indicating all my equal angles so that I know which is which, and then I know that this side here, AB is 12 centimetres.

I know that BC is Y centimetres.

And I know that AC is 10 centimetres.

So now I need to find the scale factor of enlargement.

So to do that, I need to have a corresponding side on both diagrams. You can see that here, I've got eight centimetres and 10 centimetres.

So, what have I multiplied eight by, to get 10? So we know from our work on inverse operations, that to find the scale factor, we would divide 10 by eight, which means that our scale factor is five over four.

Or if you prefer to work in decimals, 1.

25.

Now we found our scale factor.

We can have a go at working out our missing lengths.

So let's first of all, work out Y.

The corresponding side on triangle CDE is four centimetres, and we're trying to work out what Y is.

So we know, that four multiplied by our scale factor of 1.

25 gives us Y.

So we can work that out, four times 1.

25 equals five.

So Y is five centimetres.

Finally, we need to work out X.

So X is the base of triangle CDE, and the corresponding side is AB on triangle, ABC.

So we know, that X, multiplied by our scale factor of 1.

25 is 12.

And we know from our work on inverse operations, that X is going to be equal to 12 divided by 1.

25.

So X, 9.

6 centimetres.

So we've answered that question now, let's have a look at another one together.

So here we are.

We're asked, are triangles ADE and ABC similar? Pause the video now and have a go at answering this question.

When you're ready to go through it, resume the video.

So I'm just going to talk through this one then.

So we're looking for the corresponding angles in both triangles to be equal.

So, first of all, we can say that angle ADE is equal to BAC because it's a shared angle.

It's in both triangles.

Then we can look at angle ADE and angle ABC, and we can say that those are equal as well, because those are corresponding angles.

We have a pair of parallel lines again.

So that's how we know that those are corresponding.

And equally, we have, another pair of corresponding angles here.

AED is corresponding to ACB.

So we know that all our corresponding angles in the triangles are equal, so therefore they must be similar.

Then we're asked to find the unknown lengths.

So I'm going to redraw these triangles to help us figure that out.

So, first of all, triangle ADE it's A, D, E, we know that this is eight centimetres.

We know that this is 4.

2 centimetres, and we know that this is five centimetres.

Then we have triangle ABC.

So A, B, C.

And we know that this side is 14 centimetres because it's the eight centimetres from A to D and the six centimetres from D to B.

And we're not sure what this side is, and we're not sure what this side is.

So they're the two that we need to work out.

First of all, we need to find our scale factor.

So eight multiplied by what gives us 14? And again, we'll use our inverse operations.

So our scale factor is 14 divided by eight, which gives us either seven over four, if you like working with fractions, or 1.

75 if you prefer to work with decimals.

Now we need to work out our missing lengths.

So 4.

2 multiplied by 1.

75 gives us 7.

35 centimetres and five multiplied by 1.

75 gives us 8.

75 centimetres.

So we found all the missing sides in this triangle now.

You're now going to pause the video and have a go at the independent task.

When you're ready to go through some answers, resume the video.

Well done, having a go at that independent task, let's go through the answers now.

So explain why triangle ABC is similar to ADE.

Here's your explanation.

Pause the video now, and just double check that what you've written is the same as what I've written.

If you've worded it slightly differently that's okay, so long as the concept is the same.

And for length, CE your answer is 16 centimetres.

Question two, look at the diagram, explain why triangle MNO is similar to ABO.

So here's the explanation.

Pause the video now, and double check that yours is the same.

The length of OM is eight centimetres.

And the length of OB is 3.

5 centimetres.

Really well done on that task.

I hope you learned lots with that.

So now we're moving on to the explore task.

Pause the video now and have a go at this.

When you're ready to talk about some of it, resume the video.

Okay, so hopefully you've had a really good go at that task.

Let's have a look at some of the other vertices then.

What's the common vertice to all these triangles? Good, it's this one here, 00.

So that one is a common vertex of all triangles.

What about the others? What's this one here? Well, we know that this point here is 3.

472.

So this one must be 3.

470.

What about this one here? What's that? What about this one here? What have you figured out? What would happen if we increase the radius of the larger circle? What happens then? When you've drawn this diagram, what have you noticed? What are the angles? All the triangles similar? I'm going to leave that with you there.

Well done for today's lesson.

That's the end.

I hope you've learned loads and I hope you've enjoyed this lesson on nested triangles.

Just a reminder to do the quiz please, so that you can show me what you've learned.

And I really look forward to seeing you again in the future sometime.

Thanks for your hard work.

Bye.