Lesson video

Hello, and welcome to this lesson on impossible triangles with me, Miss Oreyomi.

For today's lesson, you'll been needing your paper or your book.

You would also be needing a pair of compasses and also a pencil.

So if you need to pause the video now to go get this equipment, then please do so.

And then when you're ready, come back and we'll resume the lesson.

In today's lesson, you will be able to determine when it is impossible to construct a triangle, given three lengths.

Again, just a reminder, that you will be needing your ruler, your compasses and your pencil for today's lesson.

So please do go get those equipments before you proceed with your first task.

Your first task is to identify the dots that when you connect them could give you, a, an equilateral triangle, b a triangle with side lengths of three centimetre, three centimetre, and four centimetre.

And then what other dots could you connect that would give you an isosceles triangle? So the screen now and have a go at that.

And once you're done resume the video and we can go over some possible solutions.

Let's go through some possible solutions bearing in mind that I don't have a ruler.

I'm not drawing with a ruler on my screen.

It's probably not going to look perfect on your screen.

So a possible solution, the dots I could connect, to form an equilateral triangle would be to go from the red to the purple, and then from the purple to the grey, and from the grey back to the red.

This is why you draw with a pencil and a ruler because of things like that.

If I want to connect a length of three centimetre, three centimetre and four centimetre, I'm going to change the colour just so you can see it differently.

I could start from my blue, over here, and then go to my green, and then connect it to my red and go back here.

I've got three centimetre, three centimetre, and then four centimetre here.

So these two would be equal.

And some other possible isosceles triangle, let's go for purple is time, I could go from red here to the blue over here, and then to the grey here.

So this length, this length, will be the same.

Like I said, this is why you use a pencil and a ruler, when you're drawing your triangles.

But these are possible solutions that you could have come up with.

What this slide is asking you to do is to cut out strips of paper.

One sheet of paper measuring eight centimetre, another sheet of paper measuring four centimetre, and another sheet of paper measuring three centimetre.

Can you put this together to construct a triangle? So using your paper strips, can you construct a triangle? And then where you can measure, where you can get your strips is to measure paper, using a ruler.

So take a piece of paper, measure eight centimetre, take another piece of paper, measure four centimetre, take another piece of paper, measure three centimetre.

Can you construct a triangle using those strips? And the question is, is it possible to complete the construction? Why or why not? So if it's possible to complete the construction, why? And if it's not possible to complete the construction, why is it not? So pause your screen now and attempt to construct your triangle.

Once you've had a go at that, resume the video and we'll see whether it's possible or not to complete the construction.

So after cutting up my sheets of paper, this was the image I was able to come up with.

So I could not construct a triangle from having eight centimetre paper strip, four centimetre paper strip, and three centimetre paper strip.

And we are going to find out why I couldn't construct such triangle.

We're still trying to construct this triangle, but now we're moving away from using paper strips to trying to construct this using our compass.

So in your books, can you construct this triangle in your book? So you can choose any of this length as the base length.

And then try to construct the remaining two side lengths using the length as the radius for the circle.

So once you've done that, there are three questions I want you to be thinking about.

The first one, what do you notice about the circles? Secondly, where is the intersection happening? And then thirdly, what about the side lengths? So the ultimate question is, is it possible to complete this construction? If it is, why? If it's not, why not? So pause the video now, attempt this, once you're ready, come back and then we'll go through our observations together.

What did you come up with? Were you able to construct that triangle by using your compass? I am going to show you the video of what I did and the result I got, when I did this task.

So I'm going to talk over the video, so do pay attention.

So I decided to use eight centimetre as my base length.

Now I could have chosen any of the length.

I could have used four centimetre or three centimetre, but I chose eight centimetre.

And then as usual, I measured three centimetre, so that I could draw my radius circle of three centimetre, rather draw my circle with a radius of three centimetre.

So that's one circle drawn.

Now the other length is obviously four centimetre.

So I am measuring four centimetre using my compass and my ruler.

And then on the other side, I am going to draw a circle with a radius of four centimetre.

I'm going to pause the video just here.

What are we noticing about the circle? So first question, what are we noticing about the circles? They don't meet, do they? They don't touch.

They're not close to each other.

Also where's the intersection? Is there any intersection happening? Can we see any intersections here? Well, no, the circles don't intersect.

Therefore we can't draw the side lengths.

We can't draw the side lengths because the circles don't intersect.

And previously we learned that when we have a given lengths, three given lengths, we draw the triangles, they meet where the circles intersect.

So this is an impossible triangle because the circles do not intersect.

So if we go back to the picture, this is a visual representation of an impossible triangle, because the circles do not intersect, we can't actually construct this triangle.

Nor could our paper strips be used to make these triangles because of what the video just showed us, that they don't intersect.

And again, this is just another way of, this is just another way of saying that the circles don't intersect, therefore the triangle cannot be formed.

So we've determined that there are some triangles that are impossible to construct.

But when are triangles impossible to construct? Let's take a look at this question.

Two sides of a triangle have a length seven centimetre and five centimetre.

What could the other length be? Previously, in our first activity, we saw that my base length was eight centimetre and my two side lengths were four and three centimetre.

If I add my two shorter sides together, that gives me seven centimetre.

But because seven centimetre is less than eight centimetre, I could not construct my triangle.

Therefore, my third length must be greater than my two shorter sides.

So for example, if my length here, if I say the length here is two centimetre.

The two shorter sides is five and two.

Five and two is seven.

So say for example, I try to construct this triangle here.

So this would be two centimetre, this is a rough sketch.

If I draw this circle of a radius seven centimetre, and then I come here.

Let's do that again.

I've got two centimetre here.

If I draw this circle and it has a radius of seven centimetre, and I come to this point, and I draw another circle with a radius of a five centimetre, this seven centimetre and this five centimetres, this seven centimetre radius circle, and this five centimetre radius circle, they would just touch.

There would be no intersection.

Therefore, since there was no intersection, we can't construct a triangle.

So we can say that, missing length here must be, our missing length here must be more than two centimetre because if it's two centimetre, that's just seven, five plus two is seven.

I need my value to be greater than seven.

So if it would be 2.

1 is fine, 2.

2 is fine, three is fine, four is fine.

The value must be between two and 12.

Why must it be between two and 12? So my value for my length must be between two and 12.

My values must be between seven takeaway five, which is two and five add seven, which is 12.

So my value must be between the two smallest, taking away the two values and adding the two values I've been given.

So the question is, can I write this as an equality? If I've been given this, can I write this as an equality? I could say the length L would be greater than two, but also less than 12.

What if you have these three examples here, eight centimetre and five centimetre, a triangle with nine centimetre and four centimetre, and a triangle with a centimetre and b centimetre.

What could the third length possibly be for each of these triangle? Pause your screen now and attempt it.

Write the possible values for this.

And then once you're done, we'll resume and look at possible solutions.

We have a triangle with eight centimetre and five centimetre, and we want to find what the third side length would be.

The third side length would have to lie between three centimetre and 13 centimetre, because if it's three centimetres, five plus three is eight, eight is not greater than eight.

So it's got to be more than three.

So it could be four, for example, five plus four is nine, nine is greater than eight.

If our value is 12, we would add eight and five together, 13 because they will be the shortest value in that triangle.

And 13 will be greater than 12.

For nine and four, our values must lie between a five and 13, because six, for example, plus four is 10 and 10 is greater than nine.

And again, if we choose a value of 11, nine plus four is 13 and that is greater than 11.

So we're looking for values that if I add the two shorter sides together, it would always be bigger than the largest side.

And then if we're thinking of a way to make it more general.

So if I don't have a specific value, if my value is a, and then my second length is b, assuming that the value of a is greater than b, then I know that I would subtract a and b to get my smallest possible value that it could be, and then add a and b to get the largest possible values that it could be.

So by subtracting a and b an adding a and b, I could find the third value because it would lie between those two values.

Very quickly, can you identify which triangles are impossible to construct? So if you, for example, construct these triangles, which would you find out that are impossible to construct? Take 10 seconds to figure this out.

And then once you're ready, press play to resume the video.

So the triangles that have been ticked, so this one, and this triangle would be possible to construct because I've got, for example, nine and 10 is 19 and 19 is greater than 11.

Whereas this one, my two shortest value is four and two, four plus two is six and six is less than seven.

So I cannot construct that.

And this one too, one plus one is two.

Those are my two shortest value and two is less than four, so I cannot construct that.

Whereas if we go over here, three plus three is six and six is greater than three, so I can construct it.

It is now time for your independent task.

I want you to pause the video now and attempt all the questions on your screen.

Once you're done, press play to resume, and we'll go through the answers.

I hope the task was challenging and interesting enough for you.

Let's look at the answer for number one.

What are the side lengths for each of the triangles on the centimetre grid? And what do they have in common? The answers are up on your screen and what they have in common is at least two sides are equal and there are at least two equal angles.

Second question, explain why an isosceles triangle with side lengths three centimetre and three centimetre and six centimetre is impossible to construct.

If I've got three centimetre and three centimetre, the circles would not intersect.

They would literally just touch.

And therefore the triangle is area-less.

I cannot construct this triangle because there is no intersection for my two circles.

Third question, which of these triangles are possible to construct? And the answers are on your screen.

The first one is impossible to construct because three plus four is seven and seven is less than 10.

So I cannot construct this.

Whereas the second one, four plus four, my two shortest value is eight and eight is greater than six.

Also the last one, six plus two is eight, eight is equal to eight, so I cannot construct this triangle.

Let's think about this.

An isosceles triangle has a side lengths a centimetre and b centimetre.

What range of values can b take for the triangle to be possible? So I've written here that b must be greater than a.

So assuming a is two, b must be greater than a, for this to be possible.

Because two plus two is four and four is greater than three.

So I will be able to construct that triangle, but b must also be less than twice of a, 'cause two times two is four and my b must be less than four for this triangle, for me to be able to construct this triangle.

Moving on to your explore task.

There's a lot of writing here, but I'm going to read it out and break it down for you, so that it's possible for you to answer it.

Two goats in a field are tethered to opposite ends of a fence using rope.

One rope was X centimetre long, so that would be this one here.

And the other was nine centimetres long, so that would be this one here.

The fence itself is 12 metres long, and I've just realised, I've read the units wrong.

So one rope is X metres long and the other rope was nine metres long.

Suggest a value for X where the goats' roaming zone do not overlap.

So what value can X be so that the circles do not overlap? Secondly, suggest a value for X where one of the goat's roaming zones is contained within the other.

And thirdly, what is the range of values, where there is a partial overlap in the goats' roaming zones.

So where there's an overlap between the two zones.

So pause your screen now and attempt this task and come back once you're done and we can go over the answers together.

Let's go over the answers together.

The first one, suggest a value for X where the goats' roaming zones do not overlap.

The value is going to be three, two or one, because if X is three, then the circles will touch.

If X is two, the circles don't meet.

And if X is one, the circles also do not meet.

Secondly, suggest a value for X where one of the goat's roaming zones is contained within the other.

That means one circle covers the other circle.

And for that to happen, X must be greater than 21.

If X is greater than 21, then I would have something like this and something like that.

What of C? What is the range of values for X where there's a partial overlap? So this is asking for an intersection, a point where they intersect.

And we know that that would be between when X is greater than three, but when X is less than 21.

We have now reached the end of today's lesson, a very big well done for carrying on straight onto the end.

If there are points in the lesson that you do not understand, please do watch it again.

Watch that part again and take your time through it so that you fully understand this concept.

And I will see you at the next lesson.