Lesson video

Hi, I'm Miss Kidd-Rossiter and I'm going to be taking today's lesson on further triangles.

It's going to build on the work that we've already done on area of triangles.

Before we get started, please make sure you're in a nice, quiet place if you're able to be, you're free from all distractions, and that you've got something to write with and something to write on.

If you need to pause the video now to get any of that sorted, then please do.

If not, let's get going.

So we're starting today's lesson with a try this activity.

Xavier and Yasmin are finding the area of the nested triangles below.

Xavier says, "I think that the area of the two triangles are the same." Yasmin says, "No, I don't think their areas can be the same!" Who do you agree with? And can you explain your answer.

Pause the video now and have a go at this task.

Excellent work! Let's see what you think then.

We know that to find the area of a triangle from our work last time, we do the base multiplied by the perpendicular height, and then we divide it by two, don't we? That's what we saw last time from our work on parallelograms. Remember, we have our parallelogram, we can split it into two triangles.

We know that for a parallelogram, we do the perpendicular height, which is there, multiplied by the base.

So for a triangle, it must be the base times the perpendicular height, and then halved, divided by two.

So, what do these lines here mean? Can you tell me? Excellent, they mean that they are the same length, don't they? So the base of the triangle on the left is five centimetres and the base of the triangle on the right is five centimetres.

So if I was to work out the area of the triangle on the left, I would be doing five multiplied by our perpendicular height, which we can see here is seven centimetres, and then dividing it by two.

So I can see that five times seven is 35, divided by two is 17 and a half.

So the area of my first triangle is 17.

5 centimetre squared.

Is that the same for my second triangle? Let's double check.

So I've still got a base of five centimetres and a height of seven centimetres, divided by two, and yes, it does give me the same answer.

So this one also has an area of 17.

5 centimetre squared.

So I don't know about you, but I agree with Xavier.

Moving on to the connect part of the lesson then.

Two parallel lines are used to construct two triangles.

Your job is to explain what's the same and what's different about the two triangles.

Explain to me why the areas of the triangles are equal, and then imagine sliding point A along the dashed line to a point that you can no longer see.

Is the area of this triangle still the same? Pause the video now and consider these questions.

Excellent, let's go through them together then.

So first of all, what's the same and what's different about these two triangles? Well the key word in all of this is parallel.

You know that the distance between two parallel lines is always the same.

So that's really key to understanding this.

The first thing that we can see that's the same is that both triangles share the same base.

So that's something that's the same.

And then, we can see that the height here must be the same as the height here because the perpendicular distance between two parallel lines never changes.

So they have the same base and they have the same height.

What does that mean then is different about these two triangles? Can you tell me? Excellent, what's different is the position of the apex here.

So here is the apex of the first triangle on the left and then here is the apex of the second triangle on the right.

So they've got the same base, the same height, but the apex of the triangle is in a different place.

So, why are the areas of these triangles equal? Well if we know that the base is the same and we know that the height, the perpendicular height, is the same, then we know that for one triangle we'll be doing the base times the perpendicular height, and dividing it by two, and for the second triangle, it's the same base and the same perpendicular height that we would multiply and divide by two.

So that's why the area of the triangles are equal.

It doesn't matter where the apex of this triangle is.

So long as it's on this parallel line, it will always have the same base and the same height.

So if we slide point A along the dashed line, either to the left or to the right, to a point that you can no longer see, is the area of this triangle still the same? Well, yes it is because it still shares the base here and the height between the parallel lines.

The perpendicular height doesn't change.

You're now going to apply today's learning to the independent task.

So pause the video here, navigate to the independent task, and when you're ready to go through some answers, resume the video.

Excellent work, let's go through the independent task then.

So first of all, work out the area of triangles A and B in this diagram.

Well we can see, can't we, that they have the same base.

That's what's meant by these lines here.

So to work out the area of the large either rectangle or parallelogram, we would do 14 multiplied by eight which means that the big triangle of A and B are 56 centimetres between them.

And then that means that triangle A and triangle B will each be 28 centimetres squared.

Triangle C is partly on top of triangle D.

They have the same base, which we can see is 16 centimetres, and the two lines are parallel, which means the height of C must be the same perpendicular height as D, which is 12 centimetres.

And then we need to work out the area of each triangle.

We could do 16 times 12 to get 192 centimetres squared, and that means that each triangle is 96 centimetres squared.

Work out the area of the triangle here.

Then, draw a different triangle with the same base as this triangle and either the same area, a larger perimeter, or a smaller area.

So first of all, the area of the triangle.

Well our base is four units.

Our height is three units.

And remember, it's our perpendicular height.

So our area would be four multiplied by three, divided by two, which gives us six units squared.

Excellent.

So then part a.

Draw a different triangle with the same base and the same area.

So we know if it's got the same base and the same area, it must have the same height.

So all the ones I'm going to show you now are just examples of what you could've drawn.

So for part a, so long as your triangle had the same base and the same height, then it is perfectly correct.

So here's one example.

Here's a second example that you could've drawn for that.

Then we are told it has the same base but a larger perimeter.

So here's an example that you could've drawn there.

This one will still have the same area.

It shares the same base but you can see as I've slid my apex along, the two sides that are not the base have got longer.

So that means that the perimeter will be larger.

And for the final one, a smaller area.

Well if it's got the same base, in order to have a smaller area, it must have a smaller perpendicular height.

So any example of a triangle that has the same base and a smaller height, for example this one, is perfect for that.

Excellent then.

So moving on to explore task.

Are these statements always true, sometimes true, or never true? And then can you draw some examples for each? So pause the video here and have a go at this task.

Excellent work! Let's go through them together then.

So first of all, for any triangle there is another with the same area but a greater perimeter.

Well this one is always true.

So let's just think about starting with a right-angled triangle.

If we keep the same base and the same perpendicular height, let's slide the apex of the triangle along, then the perimeter of the triangle will increase.

So you can see here, obviously yours would be much nicer drawn than mine with a ruler, we've got the same perpendicular height and we've kept the same base, but the perimeter has increased.

And this will always be true for any triangle that you draw.

The second one then, for any triangle there is another with the same perimeter but a smaller area.

Well in order for it to have a smaller area, we have to either decrease the height or the base.

So if we had a right-angled triangle again, let's say, and we had a base of four units, a perpendicular height of three units, and a hypotenuse of five, and if we drew a triangle that had a smaller base.

So let's say this time the base is two units but the same perpendicular height, then this will have the same perimeter because we could say that this is now six units and this is now four units.

And we've still got the same perimeter here but the area will be smaller.

So this one, again, is always true.

Excellent work! That's the end of today's lesson.

So thank you very much for all your hard work.

Don't forget to go and take the end of lesson quiz so that you can show me what you've learnt, and hopefully, I'll see you again soon.

Bye!.