Lesson video

Hello there and welcome to today's lesson.

My name is Dr.

Rowlandson and I'll be guiding you through it.

Let's get started.

Welcome to today's lesson from the unit of plans and elevations.

This lesson is called checking and securing understanding of accurate drawings.

And by the end of today's lesson we'll be able to accurately construct circles and triangles using a pair compasses and protractor respectively.

Here are some previous keywords that will be useful during today's lesson.

So you may want to pause the video if you want to remind yourself what any of these words mean and then press play when you're ready to continue.

This lesson contains two learning cycles and each cycle we're going to be drawing triangles, but to start with, we're going to be constructing triangles with a pair of compasses.

A pair of compasses can be used to draw a circle accurately.

Here's a pair of compasses in this photograph here, the pair you have may look different to this.

It may be plastic, for example, or it may contain other features.

Let's take a look at the one we have here.

We have our compasses, which two of them make a pair.

We have a handle for holding.

We have a place for you to put your pencil and we have a needle which holds the pair of compasses in place on the paper.

Let's take our pair of compasses now and place a pencil into the pencil holder.

It should not matter how close the pencil tip is to the needle, but it's easiest to draw when the pencil and needle tips are aligned.

For example, with these three images here, you can have your pencil in any of those ways and draw a circle just about, but with the middle one, it's probably the easiest that way.

That would be most advisable to have them as close together as you can.

When you are turning a pair of compasses, you may prefer to hold the handle and rotate it between your fingers.

Now this can be quite tricky to begin with and it does get easier with practise and once you've got quite good at this, it is the way it provides quite a bit of flexibility and freedom for how you can use your pair of compasses.

But it does take a bit of practise.

If you find it tricky, you may prefer to hold the compass near its needle and rotate the paper instead.

However, please avoid holding both compasses while you are drawing a circle because it can cause them to adjust themselves midway round and you won't have an accurate drawing them.

So let's take a look at how to draw a circle with a pair of compasses.

To draw a circle with a radius of four centimetres with a pair of compasses, open up the pair of compasses so that the distance between the needle and the pencil tip is four centimetres.

The needle is going to go where the centre of the circle is.

The pencil is going to draw the circumference of the circle, so the distance between the centre and the circumference is the radius.

So that's why you want to open up your pair of compasses to the length of the radius.

Then place the needle at the point where the centre of the circle will be and rest a pencil gently on the paper.

Don't push down hard because it will cause the pencil to slip or it cause a needle to slip.

Go easy with it, just draw it nice and gently and use it to draw your circle.

Let's check what we've learned.

Jacob has got his pair of compasses ready to draw a circle.

Which statement will be true? Is it A, the circumference will be six centimetres? Is it B, the diameter will be six centimetres? Or is it C, the radius will be six centimetres.

Pause video while you choose and press play when you're ready for an answer, The answer is C, the radius will be six centimetres.

It's a distance between the centre and the circumference where the needle is at the centre and the pencil will be at the circumference.

So try it yourself now, take a plain piece of paper and mark a cross somewhere near the middle of the page.

Using a pair of compasses, draw a circle with a radius of six centimetres and a cross at the centre.

Pause video while you do that now.

Okay, let's check how we've done.

So now draw two radii in your circle.

Pause video while you do that.

Now use a ruler to accurately measure the length of each radius.

And what you should do is check are both the radii of your circle six centimetres.

If you are draw another radius and check that one is as well.

Pause video while you do that and press play when you're ready to continue.

A pair of compasses can be used to accurately construct a triangle when all three length are known.

For example, here's a sketch of triangle PQR.

One side is seven centimetres, another is 10 centimetres, another is nine centimetres.

Laura says, "I thought pairs of compasses are used to draw circles.

So why would somebody use one to draw a triangle?" Laura attempts to construct the triangle accurately using only a pencil and ruler, no pair of compasses.

Let's see how Laura gets on.

She starts by mark at a point for Q and then draws a side which is 10 centimetres and marks point R.

She then takes her ruler and draws a side which is nine centimetres and marks point P.

And now she's gonna draw the third side.

Can you anticipate what might happen? She puts the ruler on and oh no, it's not seven centimetres, it's six centimetres.

Laura says, "QR and RP are the correct length, but QP is too short.

I need to find a point that is exactly seven centimetres from Q and exactly nine centimetres from R." And that's where our pair of compasses can come in and help us.

Laura constructs a triangle accurately by using a pencil, ruler and pair of compasses.

She starts in the same way.

She marks a point for Q and draws the line segment QR just 10 centimetres.

She then says, "I'll measure out seven centimetres and nine centimetres at the side of my page," like this.

She then says, "I'll open up my pair of compasses to the distance of each line segment," like so.

Now you don't necessarily need to draw these additional line segments at the side of the page.

You can just open up your pair compasses against your ruler, but some people find it quite fiddly to be using two pieces of equipment at the same time and trying to put 'em together that way.

You may find it easier then if you prefer to choose one piece of equipment at a time, so use your ruler to draw the line segments, accurately as you can, and then put that to one side and then use your pair of compasses to measure out seven centimetres that way.

That way you're just using one piece of equipment at a time.

Either way, we can do the same thing going forward.

We can place our pair of compasses so that the needle is at point Q and the pencil now is seven centimetres away from point Q and we can draw a circle.

Every point along that circle is seven centimetres from point Q.

We can then open up our pair of compasses to nine centimetres, put the needle on point R.

The pencil now is nine centimetres from point R, so when we draw our circle, every point along that circle is nine centimetres from point R.

Can you see the points, which are exactly seven centimetres from point Q and nine centimetres from point R.

They're here, the points where the circles intersect are both exactly seven centimetres from Q and nine centimetres from R.

So we can use that to draw the rest of our triangle.

We can draw a line segment going from Q to P, which is exactly seven centimetres, and our line segment going from P to R, which is exactly nine centimetres.

Now there is a nerve point where these two circles intersect.

Laura says, "I could have also used the other point.

This triangle is congruent to the previous one." Laura also says, "I didn't need to draw the full circles.

I just need to draw arcs that are big enough for them to intersect." Because that's the point you're looking for, where these two circles or these two arcs intersect.

Let's check what we've learned.

Here, we have a sketch and an accurate construction of triangle ABC.

Use this to find the values of X, Y and Z, which are the sides you can see on the sketch.

Pause video while you do this and press play when you're ready for an answer.

X is 10, Y is nine, and Z is 5.

5.

We can see that because X is the distance between A and B, which you can see in the accurate construction already.

Y is the distance between B and C.

Now that distance in the accurate construction is the radius of that circle, which is nine centimetres.

And Z is a distance between A and C.

Well that's the same as the radius of the smaller circle.

Here's one for you to try.

Use a pair of compasses and a ruler to accurately construct the triangle ABC which you can see in this sketch.

Pose the video while you do this and press play when you're ready to see what the answer should look like.

Here's what the answer should look like.

It can be in a different orientation to this and you can check your answer by measuring each side again with a ruler.

Or if you have someone nearby, you could ask them to check your answer instead.

Okay, it's over to you now for task A.

This task contains one question and here it is.

For each question, use a pair of compasses and a ruler to accurately construct a triangle with the three lengths given.

For example, in part A, construct a triangle where all three lengths are eight centimetres.

Now two of the triangles are impossible to draw and you'll see why they're impossible to draw when you try to draw them.

Or you might even spot before then.

For these questions, mark these with a cross.

Pause the video while you do this and press play when you're ready to take a look at some answers.

Let's take a look at some answers then.

For parts A, B, and C, here is how your answers should look.

Now, once again, your triangles could be in a different orientation to this.

Now these aren't necessarily exactly eight centimetres, depends on how big a screen is, but these diagrams are similar to the triangles that you should have to give you a sense of the proportions.

Use a ruler to accurately measure each side of your triangle to make sure it's correct.

And let's take a look at the next lot.

For D, we can draw that one, but for E, when we try and draw that we end up with what looks just like a straight line.

Those two circles only just meet and they meet at a point along that first line segment you draw, so E is not possible.

You can see that because six centimetres and eight centimetres, they add up to exactly 14 centimetres.

Whereas F, once you draw the longest line segment, you'll notice that the two smaller circles don't intersect at all.

That's because eight centimetres plus six centimetres is less than 16 centimetres.

Great work so far.

Now let's move on to the next part of this lesson, which is drawing triangles with a protractor and a ruler.

A pair of compasses can be used to actually construct a triangle when the length of all three sides are known.

Side, side side, S, S, S.

Now, while certain angles can be constructed accurately using a pair of compasses, it's not possible to construct every angle using a pair of compasses.

Therefore, a protractor may be needed to accurately construct triangles in cases where not all three sides are known, but in fact maybe two sides and an angle is known, or in cases were two angles and a side are known.

Those situations could be SAS side, angle, side or ASA, angle, side, angle.

Let's take a look at an example.

The sketch shows the length of two sides and one angle in the configuration SAS, side, then angle, then side of a triangle.

Aisha makes a more accurate drawing of this triangle.

Let's see what she does.

She says, "I'll start by drawing one of the sides with a known length." So she marks point R and draws a line segment that's 10 centimetres long and marks it point Q.

She's on one of the sides there, which is 10 centimetres.

She then says, I'll use my protractor to measure 43 degrees and mark a faint point where that is on the protractor, like so.

She says, "I'll align my ruler with point R and the faint point", that she's just made, "and then draw a nine centimetre line segment", like this.

And she marked that as point P.

She's now drawn two sides of her triangle with the angle in between at 43 degrees.

All she needs to do now is complete the triangle.

She says, "Finally, I'll connect points P and Q with a line segment to complete my triangle." And it doesn't matter how long that is because we're not told how long that is in our sketch.

There we go.

Aisha says, "I could have also drawn this triangle in other orientations." For example, here we've put the 43 degrees on the left because that's how it looked in the sketch, but we could have put the 43 degrees on the right instead and the nine centimetres would've been on the right as well.

Or we could have done it in this orientation or this one.

Let's check what we've learned.

Which triangle or triangles are a correct drawing of the triangle in the sketch? You may assume that all measurements are given accurately.

Pause the video while you choose an A, B, and C and press play when you're ready for answers.

The answers are A and B.

In both of those triangles, we have seven centimetres and an eight centimetres with 48 degrees in between them.

Well, that's not the case with C.

Here's one for you to try.

Draw the triangle in the sketch accurately by using a ruler and a protractor.

Pause video while you do this and then press play to see what the answer looks like.

Your answer could be in any orientation, so it might not look exactly like this.

We can check it by measuring each of your sides again with a ruler and also a protractor as well.

And for an extra check you could measure the third side, which should be 6.

2 centimetres or something close to that, to one decimal place.

Pause video while you check it and press play when you're ready to continue.

A triangle can be drawn accurately with a protractor and a ruler if we know the angles at either end of an edge that has a known length.

In other words, if we have two angles and a side in between those two angles.

For example, Sam draws triangle, MNO, where side MN is equal to seven centimetres.

Angle OMN is equal to 72 degrees.

And angle ONM is equal to 43 degrees.

Let's see what Sam does.

They start by drawing a side which is seven centimetres, and we can mark either side of that side by putting the point M and N.

They are seven centimetres apart.

And now we need to draw the angles that go either edge of that line segment.

Now we have two angles, OMN and ONM.

The middle letter in each of those angles is the point where that angle goes.

So for example, angle OMN will be at point M.

So let's put up a tractor at point M and let's draw a line segment that is at 72 degrees with side MN.

Now we don't know how long this side will be in the triangle, so it's better to draw a line segment that is longer than you need.

And you'll see why in a second.

Let's do the same now at point N.

If we put the protractor at point N and mark on 43 degrees and draw a line segment there, what we can see is that those two line segments intersect with each other and that point is what point O will be.

And we can join up our triangle this way.

Sam says, "I could have also drawn this in other orientations as well." That for example, "This triangle is congruent to the previous one." Let's check what we've learned.

Here, we have three triangles, which triangle or triangles are a correct drawing of triangle ABC where we are given this following information? AB is six centimetres, ABC, which the angle, is 48 degrees and angle CAB is 52 degrees.

And you can assume with these diagrams you can see here that all measurements are given accurately.

Pause video while you do this and press play when you're ready for an answer.

Answer is A.

In A, we can see we have a 48 degree angle and our 52 degree angle with the six centimetres in between them.

But that's not the case for B and C.

And here's one for you to try.

Accurately draw triangle ABC using a protractor and a ruler where side AB is six centimetres, angle ABC is 48 degrees and angle CAB is 52 degrees.

Pause video while you do this and press play when you're ready to see what an answer would look like.

Here's what an answer could look like.

Yours may be in a different orientation to this one.

We can check it again by measuring it with a ruler and a protractor.

And if you want some extra checks, side CA, that should be 4.

5 centimetres to the nearest one decimal place.

And BC should be 4.

8 centimetres to one decimal place and the other angle BCA should be 80 degrees.

So you can check your answer that way as well.

Okay, it's over to you now for task B.

This task contains one question and here it is.

For each question, you need to accurately draw the triangle based on the information you are given.

So you are given sometimes sides and sometimes angles.

For example, for part A, you need to draw triangle ABC where you are given the side length AB is six centimetres, the side length BC is eight centimetres and the angle ABC is 90 degrees.

Now it's up to you to decide which strategy and what equipment to use in each question.

Pause the video while you do this and press play when you're ready to take a look at what some answers should look like.

Let's take a look at some answers now.

For parts A and B and C, your triangle should look like this.

Measurements should be accurate to the nearest centimetre and degree.

Your triangles may be in a different orientation to this.

There may be a reflection of it or a rotation of it, but they should be congruent to it.

In other words, they should have the same lengths and same angles.

Pause the video while you check these against your own.

You can turn your page around or turn the page over so you can see through the paper clearly.

And compare it to ones you can see here and then press play when you're ready to see some more answers.

Let's take a look now at parts D and E.

You should produce two triangles that look like this.

They should both be equilateral triangles with eight centimetre long sides and 60 degree angles.

These two triangles are actually congruent.

And then let's take a look at part F and G.

Your answers should look something a bit like this.

You can check part F by also measuring the length of RP, which should be 11.

5 centimetres.

And you can also check part G by also measuring side SV, which should be 13.

9 centimetres and side VT, which should be 9.

1 centimetres if you've done it accurately.

Fantastic work today.

Now let's summarise what we've learned during this lesson.

A pair of compasses can be used to draw a circle with a particular radius.

Remember the distance between the needle of your compasses and also your pencil that is equal to the radius of the circle.

A ruler and protractor can be used to actually draw a triangle given enough information.

For example, an angle, a side, and an angle in that configuration.

Or a side, an angle and a side in that configuration.

A ruler and a pair of compasses can also be used to accurately draw a triangle where the side lengths are known.

You draw your first side with your ruler and use your pair of compasses to draw a circle at either edge of that side, which gives the distances to the third point.

Well done today and I hope you have a great day.