Lesson video

Hello, and welcome to another video.

In this video, we'll be looking at triangle centres.

Now for this lesson, you're going to need a pen or a pencil and a ruler and something to write on.

Okay.

Now that you have those things I'm Mr. Maseko, let's get on with today's lesson.

Firstly, try this activity.

So what you're going to need to do first is you're going to draw a triangle, make it a large triangle, draw three of them to test out these three students statements.

And all of them are making statements about how they are going to find the centre of a triangle.

Give each of the suggestions a go, and see which one sounds best.

Pause the video here and give this a go.

Okay, now that you've tried this, let's see what you've come up with.

Well, if we look, the first student Binh says that, "I would bisect all the angles." Well, how do you bisect angles again? With this last lesson, how do we bisect angles? We pick two points, on the two line segments that make the angle, and then we draw what? A line, find the midpoint of that line, and then we join that to the centre of the angle.

And there's our angle bisector.

Now, if you did this for all three angles, this is what you should have come up with for Binh.

So those are her angle bisectors, and what you notice is that they meet at a centre.

Now, Binh's suggestion finds what we call the incentre of the triangle.

So by drawing angle bisectors from each vertex of the triangle, the point where they meet in the middle is what we call that incentre of the triangle.

Now the incentre is the centre of the largest circle, see, the largest circle that can be inscribed inside a triangle.

Okay.

Now what about the next method? So Xavier says, "I would do perpendicular bisectors of each side and see where they meet." Now how did we do perpendicular bisectors again? Well, you can join the two ends of your lines.

So if you fold your triangles and join the two ends of your line segments and you'll find lines that bisect.

So your fold lines, would then be your perpendicular bisectors.

So when you fold your two ends of your line segments together, the fold line is your perpendicular bisector.

And this is what you should have come up with for Xavier.

And when you draw those perpendicular bisectors, here's what happens.

You find another centre, so they all meet at a specific point.

Now the centre that you find using perpendicular bisectors, you bisect the line segments of the triangle is what we call the circumcentre.

So we've learnt about the incentre, which is the centre of the largest circle inscribed inside the triangle, now the circumcentre is the centre of the circle that inscribes the triangle.

So the circle, that inscribes the triangle.

So each vertex of that triangle lies on the point, lies on a point on the circumference of the circle with the centre found by working out the perpendicular bisectors of the line segments of the triangle.

And the last method that we had, was Cala's method.

And Cala's methods just says, "I will find the midpoint of each side, and join these to the opposites vertices and see if they meet." So she just finds so measures the midpoints of each of the sides, and joins them to the opposite vertices and then see where they meet.

Now, that was, mine is not drawn accurately.

So what it should look like is this.

So when you find the mid-points, they should also meet at a single point.

And now that single point is called the centroid.

Now the centroid, unlike the circumcentre and the incentre, What was the incentre? The centre of the larger circle inscribed inside the triangle, the circumcentre was the centre of the circle, that inscribes the triangle, and the centroid is the point on the triangle that if you were to balance the triangle, so the balance would, the triangle would be able to balance on the tip of a pen.

So if you take your triangle, and you put a pen on the centroid, now that triangle would be able to balance at the tip of the pen, and that is the centroid.

And you find it by joining the midpoints of each line segment to the opposite vertex.

Now we will practise finding incentres, circumcentres and centroids in the next activity.

So draw three of the same triangle, like we have here and find the incentre, circumcentre and centroid.

If you need to look back in the video about how to find each of these, pause the video here and look back and then come back and do the task.

And once you're ready, pause the video here and give this task a go.

Okay, now that you've tried this, let's see what it is you have done.

Well, how did we find the incentre? What was, what do we have to do? Good.

For the incentre we had to do the angle bisectors.

So all you had to do was bisect all those angles and when you bisected all those angles, you would have found the incentre.

Next, how did you find the circumcentre? What did you find the circumcentre? What did we do? We drew perpendicular bisectors for each of those lines.

So for each of those lines, we drew perpendicular bisectors, and that there is our circumcentre.

And this one was our incentre.

And for our centroid, we found the midpoints of each of those lines and joined them to the opposite vertex and that was our centroid.

Now what were the definition of these? The incentre was centre of the largest circle inscribed in the triangle.

The circumcentre was the centre of the circle, that inscribes the triangle, and the centroid was the centre at which the triangle is able to what? Balance at the tip of a pen.

Now, if you've noticed each of the centres is fond by doing angle bisectors or perpendicular bisectors of finding midpoints of line segments.

And these are all things that we need to be really good at.

Now for this last task, I want you to draw one large triangle and for this triangle make sure it's large enough, and for this triangle, find the incentre, the circumcentre and the centroid.

Pause the video here and give this a go.

Okay.

Now that you have tried this let's see what it should look like.

Well, if you find the incentre first that's with our angle bisectors, so I'm just going to draw this inaccurately.

So I'm not using a ruler, but yours will be much neater than mine, cause you would have used a ruler and actually measured these lines properly.

And there is, that's our incentre, I'll label that "i", the next one we want to do is our circumcentre.

How do you find our circumcentre? We drew our perpendicular bisectors.

So we drew perpendicular bisectors to each of these lines and if you drew them accurately, the incentre was somewhere there.

And then if you drew perpendicular bisectors and you drew these accurately, what should have happened, is you should have noticed you'd have found your perpendicular bisector and then if you found your centroid.

And what you should have noticed is that those three centres all lie on a straight line.

So this was our circumcentre and that was all centroid.

And you should have noticed that those three centres for that same triangle lie on a straight line.

Now really well done for getting through this lesson.

And if you want to share the work that you did on finding the angle bisectors and the perpendicular bisectors and the mid points in order to find those different centres, ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.

Thank you for taking part in today's lesson.

Bye for now.