Lesson video

Hello, I'm Mrs Lashley, and I'm gonna be working with you throughout this lesson today.

So in today's lesson we're gonna be looking at constructing rhombi by using intersecting circles.

On the screen, there are keywords that you'd have used in different parts of your learning up to this point.

You might want to pause the video so that you can reread the definitions and make sure you feel comfortable with them before moving on.

We are gonna construct rhombi today, and so the definition of a rhombus is a parallelogram where all sides are the same length, and the plural of rhombus can be referred to as rhombi.

So our lesson is in two parts, and the first part is we're gonna think about constructing shapes using triangles, and then the second part is constructing the rhombi.

So we're gonna make a start with thinking about what shapes can be constructed using triangles.

Aisha has said that she can construct any triangle if she knows the three edge lengths.

Izzy thinks that that means she can construct quadrilaterals too.

So all quadrilaterals can be split into two triangles.

Here we've got an irregular quadrilateral, by drawing on the diagonal, we now have two triangles.

Here, we've got another irregular looking quadrilateral, but the diagonal cuts that into two or splits it into two triangles.

And here we have what looks to be a rectangle and if it is a rectangle, two right angled triangles.

Aisha's wondering if she can construct a scalene triangle, what quadrilaterals does that mean she can construct? Is this not too sure, so we can have a look at that.

And so if we have two copies of the triangle, how can we join them up? So here we've got two identical scalene triangles.

We know they're scalene because of the hash marks on the edges.

So we can see that there are three different edge lengths on these triangles, which makes them scalene.

So by rotating one and joining up the edge length that both had the three hash mark, Aisha notices that this is a parallelogram because the opposite edges are equal in length.

Izzy wonders, could we have put those two scalene triangles any other way? So going back to the original with them both orientated in the same way, if we flip one over and join them again, it's the longest edge of the scalene, but that doesn't matter, if we join it up on that edge where they've both got three marks, Aisha notices and remembers that this is called kite because adjacent edges are equal.

Izzy thinks that if you join other matching edges, so on both of these we joined up the edge that had the three hash marks, but Izzy believes that if you join any of the other matching edges, you're still gonna make kites and parallelograms. And she's correct, you can see them there.

The first two are the ones we've already looked at, so the parallelogram and the kite that we saw, but we get another parallelogram, another parallelogram if you rotate them and match them on their equal edges and two different kites.

So, so far from two scalene triangles, we can make quadrilaterals of parallelograms and kites.

Izzy said if it was a right angled scalene triangle, then the parallelogram would be a rectangle and the kite would be a triangle? And she's right.

So we haven't made a quadrilateral if it became a triangle, but we can see here that we can get a parallelogram, the rectangle or using the perpendicular nature of the right angled triangle, we can construct a triangle and that triangle would actually be an isosceles triangle because you've got those two equal edges along the side.

So Aisha has noticed that's a parallelogram, a rectangle or a isosceles triangle.

So quick check, two different scalene triangles.

So up till this point, we were using two congruent scalene triangles.

So two different scalene triangles with one shared edge length are joined together.

Name the shape created.

So pause the video whilst you are deciding on what that shape would be known as and then press play to check it.

I'm hoping you went for irregular quadrilateral.

So we haven't got all of the hash marks on there because there would be lots of different lengths.

They are joined by their common length, all the other edges are different.

So with that, we've got four different sized edges, we've got nothing that is marked to be parallel and we can use the grid behind to see that they aren't parallel, so this is an irregular quadrilateral.

Izzy wants to go through that all again but instead of scalene, what happens if we now use isosceles triangles? So Aisha said, "Oh, let's start with different ones this time." So the first time we decided to start with congruent scalene, this time we're gonna look at isosceles, we'll change it though, we'll use different ones like you just had in your check.

There is a common edge though, which is the same base edge.

So by connecting them along that same base edge, Izzy said, "That's another kite." So there are two ways to construct a kite.

We could construct a kite to using two identical or two congruent scalene triangles, but we could also construct a kite using two different isosceles triangles with a shared base edge.

And Aisha has identified, and she's right that there are no other quadrilaterals that can be made with those isosceles triangles.

So they're now gonna look at, okay, let's go back to them being congruent.

So if we've got two congruent isosceles triangles, are there other quadrilaterals we can construct? So here they are, here's our copies of isosceles triangles.

So if we join them along one of them edges that are equal, we get another kite 'cause adjacent edges are equal in length.

If we've rotated one but still join them up against each other on their equal edge, we get a parallelogram, because opposite edges are equal in length.

And then if we join them using the base edge, remember that these are two congruent isosceles.

So joining them up with their base edge, we get a rhombus.

So this is a rhombus, it's got, it's a parallelogram with equal edges.

So opposite edges are equal, but actually all four edges of the quadrilateral are equal.

So this is a rhombus.

So check, which of these quadrilaterals can be created with two isosceles triangles? So a trapezium, a parallelogram, a rhombus, a kite? Pause the video whilst you make a decision, and when you're ready to check, press play.

We can construct a parallelogram, we can construct a rhombus, and we can construct a kite using two triangles.

Izzy said, "Well, we haven't tried all of the options yet because we can get right angled isosceles triangles too.

So what do two of these create?" These are two congruent right angled isosceles triangles.

If we join them up using the longest edge, the base edge of the isosceles, then a square, if we connect them using one of their equal edges, we get a parallelogram again.

So a check, when two congruent isosceles triangles are joined at the base, they create a rhombus.

So we saw that earlier.

If the two triangles were equilateral instead, what quadrilateral will have been formed? Pause video whilst you think about that, you might want to sketch one out, and then when you're ready to check press play.

So a rhombus would still be created.

The only difference is that the edge that they have been joined by is also equal to the external edges, but it's still a rhombus because it still has four equal edges and none of the angles are right angular, so we wouldn't call that a square.

A task now for creating shapes from triangles.

On the screen there are eight quadrilaterals, and I would like you by considering the properties of the quadrilaterals, so use the notation on the diagrams, what types of triangles have constructed them? Pause the video whilst you're going through those eight questions, and then when you're ready for the answers, press play.

So A, B, C and D are on the screen currently.

So A was a square and it would be constructed from two congruent right angled isosceles triangle, B is a kite, but it can be known as an inverted kite, an arrowhead or a delta.

Anyway, it's been created from two congruent scalene triangles, C is a rhombus and it's being constructed by two congruent isosceles triangles.

We know it's a rhombus because the edges are all equal.

D is a kite and it's made from two isosceles triangles.

This time, because of where it's being joined, these are not congruent isosceles.

And then the final four, E was a parallelogram and it was made from two congruent scalene triangles, F is a trapezium made from two scalene triangles, G is a rhombus made from two congruent isosceles triangles, and H is a rectangle and it's made from two congruent right angled scalene triangles.

So we're now moving on to the second part of the lesson, which is gonna build on what we've just seen about creating shapes from triangles to get to the point of constructing rhombi.

So a rhombus is part of the quadrilateral family, just like a square, rectangles, trapeziums and other irregular quadrilaterals.

So what are the properties of a rhombus? Pause the video whilst you just think about what you have just seen in the first part, the definition that we've already spoken about, and see if you can recall that before we go through it.

It's got two pairs of parallel sides.

So we can see the notation on the diagram there with the feathers.

The one feather means that they are the pair and then two feathers are that they are a pair, so two pairs of parallel sides.

All of its edges are equal, so all the edges are the same length.

So we've put a hash mark on them to indicate that they are all the same length.

Opposite angles are equal in a rhombus.

So opposite angles are equal.

And again, we've got our arcs there to indicate the pairs.

It also has two lines of symmetry, and these are the diagonals of the rhombus.

So the lines of symmetry means that there are two congruent triangles in the rhombus.

And on these diagrams we've separated the two lines of symmetry.

So on the left hand diagram, you can hopefully see that the triangle sort of to the left of that line is identical to the triangle to the right of the line of symmetry and similarly with the right hand side.

And because of the edges being equal on a rhombus, then we know that these triangles that are congruent to each other are actually isosceles triangles.

And this means that we can construct any rhombus using two congruent isosceles triangles.

So first of all, you would construct an isosceles triangle.

So I've drawn my base edge of my isosceles triangle there.

I'm drawing the line that is gonna be the length of the other two edges of the isosceles triangle.

Using that to set up the paracompasses, moving your paracompasses to one end of the line segment, drawing an arc, repeating that on the other end of the line segment, because it's isosceles and because I chose to draw the equal edge with the paracompasses, it means I do not need to change the distance on my paracompasses.

Point of intersection is the third vertex, and I've constructed an isosceles triangle.

Then using the same baseline, construct the isosceles triangle underneath that line.

So we want it to be two congruent isosceles triangles.

So I'm basically constructing the reflection of the one we've just drawn.

Hopefully, my paracompasses is still set to seven, but I could always use my line, my guideline to check that, draw my arc, draw my arc and join them up, and here I've now constructed a rhombus.

The outside edges are all equal in length and I've used a line of symmetry to construct it.

Sam said, "How big does the base of the triangles need to be?" There is a link on the slides where you can click and have a look at a GeoGebra file to think about changing the base length and what that does to a rhombus.

I'm gonna go through that and you can hopefully make your own decisions, but you can also play around with it in your own time.

I've just opened up the link to a GeoGebra file where we're gonna explore the idea that Sam just asked about how long does the base edge need to be.

So I'm gonna move this blue point, and I want you to look at what is happening as I move it.

So we've just seen on the GeoGebra profile that the base length can change, and what didn't change was the edge lengths of that rhombus.

So here we're gonna go through the construction again.

The baseline is going to be extended, but the edges will stay at seven centimetres.

This time I've drawn a base length of about 9.

5, previously I drew it at about five centimetres.

We're gonna set up our paracompasses to seven centimetres by drawing a guideline first.

Then we're gonna move our paracompasses so the needle is at one end of the line segment, that is our base edge, draw our arc, repeat that on the other end of the base edge, they intersect.

Finish the construction.

That is our isosceles triangle that's got equal edges of seven centimetres.

They need to be seven seven centimetres so that our rhombus will have edges of seven centimetres.

We now repeat this using the same base edge below the line and we still get a rhombus.

So Sam says, "It doesn't seem to matter." And that's what we saw on the GeoGebra profile as well.

There are many rhombi with the edge lengths of seven centimetres, but the line of symmetry, the base edge of the isosceles being a variety of different lengths.

So a check, "There is only one unique rhombus with edge length 7 centimetres, true or false." Okay, so that is false.

Now justify your answer to why that is false.

Pause the video whilst you read through them and make your justification.

Press play and then we'll check.

So it's false, opposite angles are equal in a rhombus, but they don't have to be a fixed size.

And you can go back to that GeoGebra file to look at how the interior angles of the isosceles triangle and therefore the rhombus were equal because of the line of symmetry, which is the base edge of the isosceles triangle, but were never fixed to a specific size.

So do we need the base of the triangle to construct a rhombus? We know that that is the line of symmetry in the rhombus, which is where the two congruent triangles are connected to construct the rhombus.

So it's not the edge of the triangle that we need, but instead it is the ends of that edge that are important as they are the opposite vertices in the rhombus.

So let's construct a rhombus with edges of six centimetres and we're gonna try and do that without drawing the base edge or the line of symmetry.

So I've got two points and they are going to be opposite vertices on the rhombus.

I'm drawing my line of six centimetres, which is gonna be my edge length of my rhombus and using that to set the paracompasses.

So now I'm going to put my needle point of my paracompasses onto one of the vertices and draw an arc.

So we only need to draw an arc, but you could draw a full circle.

I don't need to change the distance, the radius of my paracompasses because it is a rhombus and all edges are equal.

So we're gonna keep that at six centimetres, but if you need to check that you've not changed your paracompasses, then that's where the guideline is really useful.

We have our two arcs with radius of six and they have intersected at two points.

When you construct a triangle, you only care about one point of intersection, but here we need both, they are our third and fourth vertex of our rhombus.

And then we use our ruler to join up those points of intersection, those four vertices to construct the rhombus.

They're all equal in length, six centimetres in this case, and we can mark them with our hash lines.

So Alex has drawn two edges of a rhombus, A, B, C, D, we can see those line segments, A, B, and then the line segment B, C and A, B, C is the angle or one of the angles in the rhombus.

He has challenged Sophia to construct the rest of it.

So he has started the rhombus, and he wants Sophia to finish the construction.

So we're gonna look at what Sophia's thought process here is for her challenge from Alex.

So she knows that B is the centre of the circle where A and C are on the circumference, and she knows this because it's a rhombus and therefore, the edge lengths are equal.

So if B is the centre of a circle, A B is equal to B C, which means they are both the radius of the same circle.

Sophia also says that that means that that is the edge length of the rhombus, and so she can set her paracompasses to this length.

Using one of them edges, she's used the AB edge, but she could have used the BC edge, that is the edge length of the rhombus that she needs to construct.

So Sophia's saying she then needs to draw an arc from A and from C of that length because she is trying to locate position D, the vertex D, which needs to be that distance from A and that distance from C.

So she's drawn an arc which is the edge length of the rhombus away and the same from C, and that point of intersection is vertex D and Alex congratulates her.

"Well done, you've completed my challenge." So here we had A, B, C given, so she knew what the edge length of the rhombus needed to be, and then she used intersecting circles to locate where there is a D point that is also that distance from A and C.

So a check similar to what Sophia just did, "A rhombus PQRS is being constructed.

Locate where vertex S will be." Pause the video whilst you decide where that vertex will be and then press play to check.

So it's the point of intersection between the two arcs is where the fourth vertex of this rhombus would be located.

So well done if you located S.

So Sophia sets Alex a similar challenge in return, however, this time she's only drawing one edge of the rhombus, ABCD.

So she is drawn edge A B.

So once again, we're gonna look at what Alex's thought process is to get through the challenge that Sophia has set him.

So if I use AB as a radius, I will know the locations of all the points that the same distance from A and one of those will be vertex D.

So again, AB is the edge length of the rhombus, and we know that all of the edges are equal on a rhombus.

So we can set the paracompasses to that edge length and draw a full circle.

And somewhere on that circumference is vertex D.

Alex continues to say, "Well, vertex C is going to be the same radius," because edges are equal in a rhombus, "From B." When we label A polygon we go round in order.

So this is why D is next to A 'cause you're gonna go A, B, then C, then D and back to A, okay? So I will draw a circle to show all the possible positions for C.

So he is gonna repeat what he just did, but using the needle pointer B as the centre of the circle.

And so now, somewhere on the circle around A is vertex D, somewhere on the circumference of the circle around B is the vertex C.

And the important thing that Alex has just remembered is because it's a rhombus, the distance between that vertex C and that vertex D needs to be the same length as AB.

It can't just be two random points, one on the circumference of the circle around A and one on the circumference of the circle around B.

Because their distance, the distance between those two points needs to be a certain length.

So he is going to choose a point for C somewhere on the circumference of that circle around B, and then draw a circle around that.

Where it intersects with the circle around A will need to be the position of D.

So here is the position of C that Alex has chosen.

The paracompasses is set to the same distance from the original edge length AB, and he draws his circle.

And what he said is, "The point at which this circle intersects with the circle around A is the position of D because then the radius is the same distance from A and from C." And so we have constructed a rhombus from only one edge using intersecting circles.

All of those circles have got the same radius because of a rhombus having equal edges.

And Sophia congratulates Alex 'cause it was quite a hard challenge and he got all the way through.

Alex wonders though, if he had chosen C to be where the two circles that are on the screen currently intersects, would it have had any additional properties.

Because remember, he chose C, C was anywhere along the circumference of that circle around B.

But what if he had chosen the point at which the two circles that are already constructed intersect? So this position here.

So remember, the next step was to draw a circle of radius AB around point C, and notice that AB are points on the circumference of that circle.

And this is because all of these edges are equal.

So D is where the circle around C intersects with the circle around A, different rhombus because we've chosen a different location for C, but is there anything specific about the point of intersection between the two circles? Sophia notices that this rhombus is created by two equilateral triangles as opposed to two isosceles triangles.

And if you look A to C, C is on the circumference of the circle around A, and so therefore A to C is equal to A to D to D to C to B to C to A to B.

All of the edges are equal including that internal diagonal.

So a check, when a rhombus is constructed using two congruent equilateral triangles, two of the interior angles will be 60 degrees, 120 degrees, 180 degrees.

Pause the video whilst you decide on the answers and then when you're ready to check press play.

There were two correct answers here.

If it's an equilateral triangle, then we know that there are 60 degree angles and opposite angles in a rhombus are equal, so there would be two 60 degree angles.

However, the construction also means that 260 degree angles are located next to each other to build 120 degree angle, and therefore there are two 120 degree angles in a rhombus that has been constructed from equilateral triangles.

So to finish this lesson, your final task, there are three questions on this slide for you to do.

So firstly, finish constructing this rhombus from the two given sides.

So this is similar to the challenge for Sophia from Alex.

Question 2 is to construct three different rhombi with side lengths of seven centimetres, and question 3 is to construct a rhombus with a 60 degree angle.

Pause the video whilst you're completing those three tasks and when you are ready for question 4, press play.

Question 4, construct this isometric cube where each face is a rhombus.

So a little bit of a hint here.

The three rhombi are congruent, therefore the angles at the centre point A are all equal.

You might need to ponder about this one for a little while, but actually it's easier than it looks.

Press pause whilst you have a go, and then when you press play, we'll go through the four answers.

So question 1, you need to finish constructing this rhombus, and the idea is that you have used the given edge to get the radius for your circles, draw a circle or an arc from both the end points, and they will intersect at the fourth vertex.

Question 2, construct three different rhombi with side lengths of seven centimetres.

And so this was really going back to the idea that the base edge of the isosceles triangles didn't matter.

You could change the base length and you would still construct a rhombus.

Here are three examples that you may have done similar ones to, but the important thing was that you had equal edges of seven centimetres.

Question 3, construct a rhombus with a 60 degree angle.

This is where the construction used in equilateral triangles was necessary, because equilateral triangles have interior angles of 60 degrees.

So instead of using an isosceles triangle to construct, you needed to construct two equilateral triangles joined by one edge.

And this actually led straight into question 4.

So question 4 was to construct this isometric cube.

You were told that the three rhombi were congruent, so they were identical, that means they had exactly the same internal angles.

The angles around the centre point A were therefore equal and angles around a point some to 360 degrees.

So dividing it by three works out that the interior angle at that point was 120 degrees, and then you could use co-interior angles being supplementary in parallel lines to get to the other angle of the rhombus would be 60 degrees, and when have we constructed a rhombus with 60 degree angles when we used equilateral triangles.

So here it was question 3, construct a rhombus with 60 degree angle three times and just using different base edges to construct them so that they were joined together.

If you remove the internal lines, you've actually constructed a regular hexagon as well.

So in summary today, in this lesson, we were constructing rhombi.

So a rhombus can be constructed using two congruent isosceles triangles joined at the base edge.

The base edge does not actually need to be drawn in the construction, you just need the endpoint to be your two vertices of the rhombus.

If a rhombus is constructed using two congruent equilateral triangles, then we know the interior angles are 60 degrees, 120 degrees, 60 degrees, and 120 degrees because an equilateral triangle have an interior angles of 60 degrees.

Well done today, and I look forward to working with you again in the future.