Lesson video

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

By the end of today's lesson, we're hoping that we can appreciate the diagonals of a rhombus.

Not only do they bisect each other, they are perpendicular to each other and they also bisect the angles.

There's keywords that you'll be familiar with from your prior learning, but you may wanna pause the video now just to refamiliarize yourself with those words as we use them during the lesson.

A key word for today's lesson is the word bisect, and to bisect means to cut or divide into two equal parts.

Another key word for today's lesson is altitude, the altitude of a triangle.

I'll let you read the definition there on the screen and we'll be using it and looking at it a little bit more in the lesson.

Our lesson has got two learning cycles, and the first one is all about isosceles triangles.

And the second part is then looking at properties of a rhombus.

We're gonna make a start on isosceles triangles.

Isosceles triangles have some very important features.

I'd like you just to think and remind yourself about what you know about isosceles triangles before I reveal them.

So isosceles triangles have two equal angles.

They also have two equal edges, and the angles that are equal are at the end or at the base of those equal edges.

Another feature of an isosceles triangle is the line of symmetry.

Do you get on with reminding yourself about some of the important features of an isosceles triangle? Isosceles triangle, ABC, and angle ABC has been labelled as 76 degrees.

Sam says, well that means that angle BCA is also 76 degrees.

So that's using the feature that there are two equal angles in an isosceles triangle and they're located at the bottom of the equal edges.

Aisha said, well we can actually calculate using the sum of interior angles that the angle BAC is 28 degrees, using the line of symmetry feature of an is isosceles triangle, Sam says that the symmetry of the triangle means that half of angle BAC is either side of the line and so the angle is bisected, remembering that bisecting means to cut or divide into two equal parts.

So the line of symmetry means the angle BAC has been bisected.

Aisha says, well also the line of symmetry splits it into two identical triangles.

We could use the word congruent there.

Here is a drawing of one of those triangles that Aisha is mentioning, one of the identical ones within the isosceles using in the line of symmetry.

So Sam says, using only triangle ABM, so that's one of these identical ones.

Then the angle BMA is 90 degrees because angles in a triangle sum to 180 degrees.

We know two of the other angles in the triangle so we can subtract to find what's left over from that 180 degrees sum.

Aisha has correctly identified that that means that the line of symmetry of an isosceles triangle is also perpendicular to the base edge.

We've just calculated that that angle is 90 degrees and when two edges or two lines are meet at 90 degrees, they are perpendicular.

So here's a check.

If the non-base angle of an isosceles triangle is 34 degrees, the line of symmetry will bisect the angle into two what angles.

So what size would the angles be when they are bisected? Pause the video whilst you work that one out and when you're ready to check, press play.

So I'm hoping that you divided it by two because that's what the word bisect means.

It's division of into two equal parts.

So 17 degrees.

Altitude is one of the key words for this lesson today.

And the altitude of a triangle is a line segment from a vertex of a triangle to the opposite edge.

It is perpendicular to that edge.

It's often referred to as dropping the perpendicular.

So as you go through your maths education and you work with maths, there are times when we need to drop the perpendicular.

And when we say we drop the perpendicular, what we're actually doing is drawing on an altitude to our diagram.

And one of the reasons is because this is the perpendicular height of the triangle.

So here are a couple of diagrams that show you altitudes of triangles.

So you can see it's a line segment that goes from a vertex to the opposite edge and is perpendicular to that opposite edge.

So that's why there's the symbol for a right angle.

This is also an example of an altitude, but when the vertex overhangs the opposite edge, then we would need to extend the direction of that edge to draw on the altitude.

On the screen though, there are also some non-examples of altitudes and the reasons that they're not altitudes, the first one is not perpendicular.

The second one, it's not from that vertex, it would go to the opposite edge.

So that's just going external.

And the last one there is a line second that has been drawn perpendicular to an edge, but it's not going to the vertex.

So it isn't an altitude.

Using this definition of an altitude, the line of symmetry in an isosceles triangle is also an altitude.

If you think back, they worked out that it was perpendicular.

So the line of symmetry is perpendicular and therefore it is an altitude.

So which of these altitudes is also the line of symmetry? Pause the video and then once you've made your decision, press play and then you can check.

Well, b.

And you're doing that a little bit by eye.

There was no other indicator on there that that was exactly central but a and c, hopefully you could dismiss because it was quite clear that they were not symmetrical either side.

So a line of symmetry can also be thought of as a line of reflection.

And if we reflect the triangle AMB across the line AM.

So if we use AM as our line of reflection, using information that we know about reflection from previous learning, then corresponding points are equidistant from the line of reflection.

A and M were points on the line of reflection.

So their distance was zero and in the reflection it's also zero, whereas point B was a distance from M, which means that M to B prime to the image of B would be equal because in a reflection the points are equdistant from the line of reflection.

So this means that the line of symmetry in an isosceles triangle cuts the base edge into two equal parts and that's the definition of bisection.

So it bisects it.

So we've seen so far that the line of symmetry is also known as an altitude because it is perpendicular is also bisecting the base edge that it meets.

So a check.

This diagram of an isosceles triangle shows some of its features.

Which features are missing from the diagram? So pause the video whilst you make your decisions and when you press play will go through the answers.

So the missing one was that the base edge is bisected by the altitude.

So the base angles are equal, we're already notated on the diagram.

The line of symmetry is an altitude was on there by using the right angle symbol and there are two equal edges.

We have the hash marks along those edges.

So the base edge being bisected was the missing feature and it's a very important feature.

Another check.

What are the missing values of A, B, and C? Pause the video whilst you use the diagram.

Use the notation to figure out the values of A, B and C.

Press play when you want to check them.

So the answers were 12 for A, because that base edge has been bisected by the line of symmetry or the altitude.

So if the distance from the altitude to a vertex was 12 centimetres, then it was gonna also be 12 centimetres in the other direction.

B was 32 because it's an isosceles triangle, they've both got one hash mark on each of them.

So we know that they are equal in edge.

So we can use the 32 there.

And C is 22.

This one you needed to calculate using either a right angle triangle with the 90 degrees and the 68 degrees and what was remaining or using the isosceles triangle and then bisecting the angle because the altitude is bisecting the angle.

So you've got two questions on isosceles triangles.

The first question is here on the screen.

So decide which of the following are isosceles triangles and if they are not isosceles triangles, state why not? So pause the video whilst you're going through those six questions and then when you're ready for question two, press play.

So question two, you need to add notation marks to the diagrams where appropriate.

So there are three triangles on the screen.

You need to add notation marks where appropriate.

Press pause and then when you're ready for the answers to both question one and this question press play.

So here we have question one.

Triangle A was an isosceles.

Triangle B was isosceles.

Triangle C was isosceles and D was also isosceles.

E was not an isosceles and the altitude does not bisect the base edge.

So 1.

8 centimetres does not equal 19 millimetres.

It equals 18 millimetres.

So because that altitude that is drawn onto that diagram is not bisecting the base edge, then it isn't a line of symmetry and therefore it's not an isosceles triangle.

F is not an isosceles triangle because there are not two equal edges.

Question two, you needed to add the notation marks to the diagrams where appropriate.

So they were all isosceles triangles and you knew that from the given information in the question.

So you needed to add on some hash marks to indicate which edges were equal to indicate that we had base edge was bisected.

You could add on the angle marks and the arcs to indicate which angles were equal to each other.

So we're now onto the second part of the lesson and this is all to do with the properties of a rhombus.

So a rhombus can be thought of as two congruent isosceles triangles joined together at the base edge.

So here we've got two congruent isosceles triangles and when we joined them together, this is actually a rhombus.

We could have used isosceles triangles that look like this and join them together.

We get a rhombus as well.

Each of those isosceles triangles is actually made from two congruent right angle triangles.

We saw that earlier in the lesson.

Therefore, the rhombus that's made from two isosceles is made from four congruent right angled triangles in total.

So here we've got one of those right angled triangles.

If we reflect it across that edge, we get our isosceles triangle.

If we reflect it along that short edge, we get one of the isosceles triangles and reflect it again.

There are four congruent right-angle triangles within a rhombus.

The dashed lines are the diagonals of the rhombus.

So what can we infer about the diagonals of a rhombus? Using the diagram, what can we infer about the diagonals? Pause the video to have a think about that.

And when you are ready to check or to go a bit further with the lesson, press play.

So we can infer that they are perpendicular to each other.

Those dashed lines have got 90 degrees symbols where they meet, where they intersect.

So they are perpendicular to each other.

They bisect each other.

If you focus on one of the diagonals when it crosses, when it intersects with the other, there is the same distance.

And we can use the hash marks here to see that those distances are the same either side and that works from both viewpoints.

They also bisect the angle that they pass through.

So the angle either side of the diagonal is equal again using the notation marks to recognise that.

So check the diagonals of a rhombus, bisect each other, are perpendicular to each other, are always equal in length, bisect the angle.

Pause the video whilst you make your choices and when you're ready to check them, press play.

So they were all correct except from C.

So they're not always equal in length, but they do always bisect each other.

They are always perpendicular to each other and they do by set the angle that they pass through.

So here's a rhombus with the diagonals drawn on and one angle given.

So they're not dashed this time, they are solid but they are the diagonals.

Alex is asking, "Is this enough information to know all other angles on this diagram?" Once again, you might wanna think about that question.

You might wanna pause the video and see how many other angles you can come up with.

So Alex has said, "well, diagonals in a rhombus are perpendicular to each other." We just saw that they are always perpendicular to each other, which means we can add on four 90 degree angles.

We are using the symbol for a right angle there.

Diagonals of a rhombus also bisect the angle that it passes through, which means that the other side of that diagonal would also be 68 degrees.

The 68 degrees is one half of the angle that it's bisected.

We know that two congruent isosceles triangles have created the rhombus, which means that base angles of the isosceles are equal.

So the other side of that rhombus would also be 68 and 68.

They have been bisected.

And Alex is very familiar, as are we.

The angles in a triangle sum to 180 degrees.

There are many triangles in this diagram.

So you could use a variety of triangles to work out the next few angles, but we'd then be able to work out that those angles were all 22 degrees.

And so Alex has said the anterior angles of this rhombus are 44, 136, 44, and 136.

And that's by totaling the bisected angles.

So the 68 at the 68 would be 136 and the 22 and the 22 would be 44.

And if you remember when we looked at the properties of a rhombus, they are opposite, opposite angles in a rhombus are equal.

So a check.

Here is a rhombus with the diagonals marked on and one angle given.

Work out the value of A, B and C.

Pause the video whilst you're working out those values and then when you're ready to check them, press play.

So a is 90 degrees, diagonals of a rhombus are perpendicular to each other.

Perpendicular means that they meet at 90 degrees.

B is 114 degrees so that 57 degrees is half of the angle b.

Opposite angles in a rhombus are equal and the diagonals bisect the angle.

So there would be 57 degrees on the other side of the diagonal, which totals to 114 degrees.

And c is 33 degrees.

There's a few ways that you could have come up with this.

You could use an isosceles triangle that base angles are equal and if you know that the third angle is 114, then you can subtract that from 180, the sum of the interior angles and then half it because they are equal.

The alternative way is you could have used a right angle triangle, one of the right angle triangles within the rhombus and you know it's a 57 and a 90 and to what is left over from the sum of the triangle.

Izzy thinks she's drawn two sides and a diagonal of the rhombus.

And there you can see her attempt of two sides and a diagonal of the rhombus.

Jun thinks that Izzy didn't manage to do it.

Who's correct? So do you think Izzy has managed to draw two sides and a diagonal of the rhombus or do you think that Jun is correct and she hasn't managed to do that? So Jun is correct.

Izzy hasn't managed to draw two sides and a diagonal of a rhombus.

The diagonal should bisect the angle that it passes through, but even without using a protractor to check whether the angles are equal, either side of it is quite clear that those two angles are not equal.

So Izzy tries again, but this time she draws two edges and both of the diagonals of the rhombus.

How can she check that this is accurate? So what checks could she do using equipment that she has done this accurately? So she could use a protractor and she could use that protractor to check that the diagonals are perpendicular to each other.

So she could check that they are all 90 degrees.

She could also check that the angle that the diagonal passes through has been bisected so that the angle either side of it are equal.

And then she could use a ruler to check that the diagonals have bisected each other.

So they are split into two equal parts.

Measure either side and are they equal.

And also the rhombus's edges need to be equal.

So the two edges that she drew, are they the same length? Because if they're not the same length then it cannot be a rhombus because the edges of a rhombus are all equal.

Check, which of these show two edges and a diagonal of a rhombus? Pause the video whilst you decide which of those have been drawn well.

And then when you're ready to check press play.

A has been drawn well, it's bisected the angle and the edges of the rhombus are equal.

B, same thing, bisected the angle and the edges are equal.

C, you can slightly see that that diagonal is not bisecting the angle, even if you don't use a protractor by eye, you can see those angles are different and that's the same for d.

For d, the diagonal is not bisecting the angle.

So if you have two perpendicular lines, can a rhombus be constructed? So this would be the point at which the diagonals intersect because we know that the diagonals of a rhombus are perpendicular to each other.

And the lines can be extended in all directions, but they need to be equal to each other on either side because the diagonals bisect each other.

So if we are constructing a rhombus, we need to make sure that the properties of the diagonals are holding.

So using a pair of compasses because that way we can get an equal distance either side, the distance doesn't matter.

Remember that the diagonals don't have to be equal to each other.

We can mark a distance away from that point of intersection and keep that consistent either side.

That means that the, that would be the length of our diagonal where the diagonal is bisected.

I'm not using the ruler there to measure, I'm using the ruler to just draw accurately a line.

I'm gonna repeat that process.

I've changed the radius of the circle that I'm drawing.

It could be equal, but I'm changing the radius here.

I've now marked the same distance either side of that point of intersection because of the idea that the diagonals bisect each other.

Using a ruler, again, not as a ruler, but instead as a straight edge drawing on that diagonal.

So here I've got the two diagonals of a rhombus.

They are perpendicular to each other and they are bisecting each other.

Joining up the ends of those line segments, we have constructed the rhombus.

We've got all of the notation on there to indicate that the edges are equal in length, our diagonals are bisected and our diagonals are perpendicular to each other.

So a check.

The diagonals of this shape are perpendicular to each other.

This must be a rhombus.

Do you agree with Aisha? So pause the video whilst you make a decision on that and when you're ready, press play to check the answer.

I'm hoping you went for no as well.

So in a rhombus, the diagonals are perpendicular, but they also need to bisect each other.

So on that construction that we just did, we had the pair of compasses to mark the equal distance away from the point of intersection.

Here you can see that this is a kite rather than a rhombus.

So the diagonals of this rhombus have lengths of 10.

5 units and 5.

8 units.

What is the area of the rhombus? So we can see the diagonals in there, inside that rhombus.

And we've been told the lengths of the diagonals.

So the rhombus is made from two congruent isosceles triangles or four congruent right-angled triangles.

And we can use this fact to calculate the area.

So if we use the two congruent isosceles triangles first, so this isosceles triangle has a base length of 10.

5, that's one of the diagonals and a perpendicular height of 2.

9 units.

And that's used in the fact that the diagonal has been bisected by the other diagonal.

So the area of a triangle is half times base times perpendicular height.

So this particular isosceles triangle has an area of 15.

225 units squared or square units.

And therefore the area of the rhombus is twice that because the rhombus is made from two congruent isosceles triangles.

So the rhombus has an area of 30.

45 square units.

If we went about it using four congruent right-angle triangles, then we would know that the base length of this right-angle triangle is 5.

25 because the diagonal has been bisected and half of 10.

5 is 5.

25 and its perpendicular height would be 2.

9 again, because half of the diagonal.

So the area of that triangle would be 7.

6125, which isn't surprising that that is half of the area of the isosceles because we know that two of those create the isosceles and therefore the rhombus has its area of 30.

45 by timing 7.

6125 by four because four of those triangles create the area of the rhombus.

So a check.

Similar to that for you to have a go at doing.

The diagonals of this rhombus have a length of 7.

5 centimetres and 9.

4 centimetres.

What is the area of the rhombus? Pause the video whilst you work that area out and when you are ready to check it, press play.

So the answer is 35.

25 square centimetres.

What you may have done to get there, there are three sort of options for the calculation you did.

So there were two which were timing by two, and that was using the two different isosceles triangles.

And there was one by using the four congruent right-angle triangles.

So check your calculation, you only needed to do one of them.

The answer was 35.

25 square centimetres.

So you're now gonna do some practise to do with the properties of a rhombus.

So question one is on the screen and you need to work out the missing angles in these rhombi.

Pause the video whilst you're working those out.

And then when you're ready for the next question, press play.

Question two is all about working out the area.

You need to work out the area of each rhombus in each part.

Pause the video whilst you're working that out, remember your units and when you're ready for question three, press play.

Question three is constructing the rhombus from this set of perpendicular lines.

So remembering the properties of the, especially about the diagonals of a rhombus to be able to construct.

Pause the video whilst you're doing that.

And when you're ready, we'll go back through the answers to both question one and two as well as this one.

So question one, you need to work out the missing angles and they were 110, 90 and 35 on rhombus A.

And then they were 16 and 74 on rhombus B.

Using the fact that the diagonals are perpendicular to each other, they bisect the angle that they pass through and opposite angles in a rhombus are equal, you should have been able to get to those values.

Question two, the area for a was 24 square centimetres.

There is an example of the calculation you may have done to get that answer.

Part b was 20.

25 or 20 and a quarter square centimetres.

Again, there is an example of how you may have come to that answer.

And question c, this one was slightly different in the wordings.

Hopefully, you read that carefully enough.

You were told the length of half of the one of the diagonals, so one of them had already sort of been bisected.

The area was five square centimetres.

And again, there's an example of how you may have calculated it.

Question three.

This is an example of a rhombus that you may have constructed.

The important thing was that you had marked an equal distance above and below the point of intersection and to the left and to the right of the intersection so that the diagonals were bisecting each other as well as being perpendicular.

At that point, you could then connect the four end points to construct your rhombus.

So in summary, the diagonals of a rhombus are perpendicular to each other.

They bisect each other and they bisect the angles that it passes through.

And there is a complete diagram of a rhombus showing all of the properties use the equal edges, the opposite angles are equal, the angles are bisected, the diagonals are perpendicular and they bisect each other.

Well done with this lesson today.

I look forward to working with you again in the future.