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 Loci and Constructions.

This lesson is called, "Construct a triangle given three side lengths using compasses." And by the end of today's lesson, we'll be able to construct a triangle given the three required side lengths.

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 is broken into two learning cycles.

In the first learning cycle, we're going to be drawing triangles with dynamic software and that will allow us to explore the theory behind the constructions and how the triangles are drawn, and drawn in a very accurate, pure way.

And that means, when it comes to learning cycle two, we can transfer what we've learned into drawing triangles with apparatus such as a pair of compasses.

Let's start off with drawing triangles with dynamic software.

Here we've got Laura who attempts to draw a triangle with side lengths five centimetres, four centimetres and six centimetres, using only a pencil and a ruler.

Now you might think that sounds like quite an easy task.

You draw a line segment which is five centimetres long, and then another one which is four centimetres long, and then you draw another one which is six centimetres long.

But let's see what goes wrong when we do this.

Laura draws the first side, which is five centimetres long.

She then draws another side, which is four centimetres long, and then she goes to connect them up to draw the third side.

And oh no, it's too short.

Laura says, "Two sides are the correct lengths, but the third side is incorrect." When Laura draws the first side, it creates two points and that means she needs to find a third point that is exactly four centimetres from one edge of the first side and three centimetres from its other edge.

So how do we go about doing that? Laura says, "Drawing circles could help me.

Circles show all the points that are a given distance from its centre.

Therefore, each circle would show all possible points where I could possibly draw my sides to." We could explore how to do this accurately using dynamic software and then consider how to replicate the same process by hand.

So let's do that.

Dynamic geometry software can be used to carry out constructions quickly and accurately.

These constructions can be used to draw triangles based on information given, such as the three side lengths.

It's called dynamic software because objects can then be moved and manipulated after a shape has been created.

And the constructions used in dynamic software can be replicated, on paper, using a pencil and apparatus such as a pair of compasses.

The dynamic software we're going to use is going to be GeoGebra.

So to get started with GeoGebra, open up a web browser and go to GeoGebra.

org.

Then find and press the button that says `start calculator' and that'll open up a blank page.

Find and press the `graphing' button to open up the options menu.

And what you wanna choose is the `geometry' one.

And then you'll see a page that looks a bit like this.

You've got a blank canvas on the right and some tools on the left.

You want more tools than that, so click `more' to access more tools and then you can see a few more appear, but there are even more tools than that.

So scroll down to see the full range of tools available, but then click `more' again to access even more tools if you do need them.

So, let's draw a triangle with side lengths five units, four units and six units using GeoGebra.

To draw the first side, which is five units, find the section of tools under the subheading of `lines,' then click on `segment with given length.

' Click somewhere on the canvas to plot your `first point' and then type in the length of the side and press enter.

In this case we're going to type in five.

And that'll create a line segment, looks a little bit like this, with two points on.

Those will be two points on our triangle.

Points A and B are exactly five units apart and we can move either of the two points around, but they will always remain the same distance apart at all times if you used that line segment given length tool.

Now let's look at drawing this second side.

Well, there are lots of options for that.

It's four units long but it could go in different directions.

So to find options for the second side, find a section of tools under the subheading `circles.

' And then click, `circle, centre and radius.

' Click on one of the points, what you've already plotted, and then type in the length of the second side and press enter.

So in this case we'll type in four for four units.

Now plot a circle like this with a centre at one of the points of our triangle.

Every point around that circle is four units from point A.

So our third point of our triangle could be anywhere on that circle, but it'll still have to be six units away from the other points on our triangle.

Now we could plot any point on this circle to draw a triangle like this, and if we did that, two of our sides will definitely be the correct length.

One of them will be five units.

That's the first one we drew.

And either one will be four units because that's the radius of the circle.

But the third side may not necessarily be the correct length.

We need to make sure we find a point somewhere on that circle that's exactly six units away from the second point that we created.

Point B in this case.

So to find options for the third side, we're going to do the same process again.

We're going to find a section of tools under `circles' and click, `circle: centre and radius.

' Click on the unused point, type in the length of the third side, which is six, and press enter.

And that plots a circle with a radius of six units.

So let's review what we've got so far.

The two centres of these circles are five units apart.

Every point on this circle is exactly six units from its centre.

Every point on this circle is exactly four units from its centre.

So where could the third point of this triangle go? Pause the video while you think about that and press play when you're ready to continue.

Well, what we are looking for is where these two circles intersect.

So to find options for the third point, find a section of tools under the subheading `point' and click `intersect,' and then click on each of the two circles and that will show the points where they intersect.

Each intersection is exactly four units from the centre of one circle and six units from the centre of the other circle.

And we know that the line segment between those two centres is five units.

So either of those two intersection points can be used with points A and B to create the triangle.

And the two possible triangles would be congruent.

So to complete a triangle, find a section of tools under the subheading `polygons' and click `polygon.

' Click on the centre of each circle and on either of the points where they intersect.

For example, the top one in this case.

And that will create a polygon that looks a bit like this.

And this is a triangle.

It is a triangle with side lengths five units, four units and six units.

Let's see where they are.

Five units is a distance between the two centre points, the radius of one of the circles is four units, and the radius of the other circle is six units.

Once you've drawn it, you can manipulate a triangle by moving any of the points, but if you've constructed it the way we've just done, then all the points will always remain the same distance apart.

So each new triangle that you create will be congruent to the original one.

So let's check what we've learned.

Here we've got Andeep who is using GeoGebra to accurately draw a triangle with side lengths five units, four units and three units.

He's drawn this first side, but which tool did Andeep use to draw the first side? Pause the video while you choose either A, B or C and press play when you're ready for an answer.

The answer is D.

He used a segment with given length tool And now he's drawn two circles.

Which tool did Andeep use to draw each circle? Pause the video while you choose and press play when you're ready for an answer.

The answer is A.

He used a circle: centre and radius tool to draw each one and that allows him to select what the radius is.

And then he's plotted points C and D.

Which tool did Andeep use to do that? Pause the video while you choose and press play when you're ready for an answer.

The answer is B.

He used the intersect tool because those points show where those two circles intersect.

True or false? The top intersection, point C, should always be used to construct the triangle.

Is that true or is it false? And justify your answer with one of the points below.

Pause the video while you do it and press play when you're ready for answers.

The answer is false because you can use either intersection.

The two triangles would be congruent.

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

This task contains two questions and here is question one.

Open up a web browser, go to GeoGebra.

org and open up a blank geometry page.

And then follow the instructions you can see on the screen here.

Pause the video, while you do that and press play when you're ready for question two.

And here is question two.

Open up your web browser, go to GeoGebra.

org and open up a new blank geometry page.

And then you've got some triangles to draw in part A to E and some questions to consider for parts F to I.

Pause the video while you do that and press play when you're ready for answers.

Okay, let's go through some answers.

For question one, for part A, you need to draw a line segment that has a length of 10 units and then use your line segment from part A to draw a triangle with side lengths 10 units, eight units and seven units.

It should look something a bit like this and here is it with the lengths drawn on.

Now your triangle might look like this one or it might be in a different orientation.

So let's do some extra checks to make sure it's right.

Under `measure,' click on `angles' and then click on your triangle and it'll show the angles on a triangle.

And if you've plotted it correctly, your angle should be 44 degrees, 83.

3 degrees and 52.

6 degrees for this particular triangle.

And as an extra check, under `measure,' you could click on `area' and then click on your triangle and it'll show the area of it.

And if you've plotted it correctly, this triangle should be 27.

8 square units.

For part C, you have to, somewhere else on the page, draw a line segment that had length eight units and then use your line segment from part C to draw a triangle with lengths eight units, seven units and 10 units.

It should look something a bit like this.

And here it is with your measurements drawn on.

Some extra checks, you could do the same again.

Under `measure,' click on `angles' and that should show these angles here.

Under `measure,' click `area' and that should show that area there as well.

And then part E said, "Explain why the two triangles you have produced in parts B and D are congruent." Well all three sides of the triangle are the same length, so they are congruent based on the rule of side, side, side.

However you have also just seen how to show the angles.

So alternative justifications can be given using any angles found as well, such as the side, an angle and a side.

In question two, you have to draw a triangle with side length six units, six units and six units.

It would look like a bit like this or it could be in a different orientation or yours might be zoomed in or zoomed out.

That's fine.

For your extra checks, your angles should all be 60 degrees and the area should be 15.

6 square units.

For part B, your triangle should look something a bit like this or in a different orientation.

For your extra checks, two angles should be 68 degrees and one should be 44 degrees and the area should be 22.

2 square units.

Part C should look something like this or in a different orientation.

In terms of your extra checks, you should have these angles here and an area of 15 square units.

And part D, your triangle should look something like this or a different orientation.

So your extra checks would be to show these angles, and again, 15 square units.

And part E, your triangle should look something a bit like this.

And for your extra checks, you would have 68 degrees, 68 degrees and 44 degrees and an area of 50.

1 square units.

And then you use those triangles to answer these questions.

Which previous question produced an equilateral triangle? That'd be part A.

All the side lengths were the same.

Which previous question produced an isosceles triangle? That was part B and part E.

In those two side lengths are the same and the other one was different.

Which previous questions produced triangles that are congruent to each other? That was part C and part D.

They had the same three side lengths each time.

And in I, which previous questions produce triangles that are similar to each other? That is parts B and parts E.

Those did not have the same side lengths but they were in proportion to each other and also according to our extra checks, they were also had the same angles as well.

Great work so far.

Now let's consider how we can replicate what we've just done, but using apparatus such as a pair of compasses.

Here we have a diagram that shows a triangle that was constructed using dynamic geometry software.

It has side lengths of five units, four units and six units.

And here we have Sofia.

Sofia wants to accurately draw the triangle on paper using a pencil and some other appropriate equipment.

And she says, "I have a pencil, ruler and a pair of compasses." How could Sofia use her equipment to draw the triangle accurately? Think about the things that are involved in the construction that you can see on the screen on the left and how you might replicate those using this equipment.

Pause the video while you think about it and press play when you're ready to continue.

Let's take a look together.

A pencil, ruler and pair of compasses can be used to accurately draw a triangle with side lengths of five centimetres, four centimetres and six centimetres.

And let's now do it together.

You can use a ruler to measure and draw the first side.

So we can draw a side length of five centimetres.

It doesn't really matter what direction that first side length goes in, so that's why we can use the ruler to draw it, just in the way we have.

Now let's think about the second side length, which is four centimetres, but we don't necessarily know yet which direction it would go in.

So we wanna draw some options.

Open up your pair of compasses to four centimetres apart.

Place the needle of your compasses at one of the edges of the line segment and draw a circle, like so.

Every point on this circle is exactly four centimetres from its centre.

So that means you could plot a point somewhere on this circle to create a triangle, a bit like this.

And two of the side lengths will definitely be the correct lengths.

The five centimetre one will be because you use a rule to draw that.

And the fourth centimetre length, that'll be correct because that's the radius of the circle.

But the third side will not necessarily be the correct length.

So after we get to this stage, we've got two points on our triangle, but the third point needs to be four centimetres from the point on the left and also six centimetres from the point on the right.

Now we know that anywhere on the circle will be four centimetres from the point on the left, but we need to find somewhere that is six centimetres from the point on the right and we'll do that by drawing another circle.

If we open up our pair of compasses to six centimetres apart, place the needle of the compasses at the other edge of the line segment and draw a circle.

It looks something a bit like this.

In my case, the circle's gone off the screen.

Now that's not a problem 'cause you don't necessarily need to draw a full circle.

An arc can be sufficient so long as it intersects your first circle at least once.

So let's review what we can see so far.

Points A and B are five centimetres apart.

Every point on this arc here is exactly six centimetres from point B.

And every point on this circle is exactly four centimetres from point A.

So where could the third point of our triangle go? Pause the video while you think about it and press play when you're ready to continue.

Well, we have these two points here, which is where the arc and circle intersect.

These points are both four centimetres from point A and six centimetres from point B.

Therefore we could choose any intersection to be the third point to complete our triangle.

Like here, we can see on the screen.

we could have used the other intersection instead.

And if we had, all three side lengths would still remain the same and this triangle would be congruent to the previous one.

Now we don't necessarily need to draw full circles or such big arcs when we do our constructions.

Rather than drawing the full circles or large arcs, smaller arcs could be used instead.

The arcs only need to be long enough so that they intercept at least once.

However, it can be difficult to visualise where they will intersect.

So you may wish to draw longer arcs if you're unsure.

The more you practise this, the more you get a sense of where the third point would be.

But maybe to begin with, draw quite long arcs so you know they definitely will intersect.

And then you can always make 'em shorter and shorter as you get more used to things.

If you're ever unsure, you could always draw the full circle and that's absolutely fine.

So let's check what we've learned.

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

You've got lengths labelled on the accurate drawing, but on the sketch, the sides of the triangle are labelled X, Y, and Z.

Please find the values of X, Y, and Z.

Pause the video while you do it and press play for answers.

Here are the answers.

X equals seven, y equals eight and Z equals six.

Okay, try one yourself now.

Use a pair of compasses and a ruler to make an accurate drawing of this triangle you can see in a sketch here, triangle ABC.

Now constructions can always be tricky the first time you do it, so be patient with it and don't worry if you don't get it perfect first time.

Pause the video while you have a go and press play when you're ready to see an answer.

Here's what your answer could look like, with a pair of arcs on to show where the points may intersect.

Now yours might look like that one, it might be a different orientation.

Your arcs might be at a different point on the triangle if you chose to draw a different side first.

But you should have a triangle that looks something like this when you turn it around.

You can check your answer by measuring each side again with a ruler and making sure they're accurate.

Or if you've got a partner nearby, you could ask them to measure your triangle instead.

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

This task contains three questions.

And here is question one.

Take a sheet of plain paper, a pencil, a ruler and a pair of compasses and follow the instructions on the screen here to construct some triangles.

Pause the video while you do it and press play when you're ready for question two.

And here is question two.

Once again, take a plain piece of paper, a pencil, a ruler and a pair of compasses.

Part A asks you to construct an equilateral triangle.

And then part B, asks you to pick any of your sides and construct a perpendicular bisector for that side.

And then, part C, consider why does your construction from part A cause you to bisect an angle? And then part D, you apply some Pythagoras's theorem here to calculate the shortest distance from a vertex to its opposite side.

And then you can use your ruler afterwards to check how accurate that is on your diagram.

Pause the video while you do this and press play when you're ready for question three.

Here is question three.

The diagram shows the sketch of a triangle and what you need to do is make an accurate drawing of a triangle that has a side length of six centimetres and is similar to a triangle represented in the sketch.

Now there are multiple possible answers for this, so create one and then you can maybe think about what other ones you could do afterwards.

Pause the video while you do it and press play when you're ready to see some answers.

Let's take a look at some answers.

For question one, parts A and B led you to construct this triangle here.

Now yours might be in a different orientation to this and that's absolutely fine.

As an extra check, you could use a protractor to measure the angles and they should be 44 degrees, 83 degrees and 53 degrees.

If you did part one of this lesson, you may notice that this triangle is congruent to one of the ones you constructed earlier on dynamic geometry software.

You could hold this triangle up towards that one and make a comparison that way as well.

Parts C and D should lead you to construct this triangle here.

And again as an extra check you can measure the angles and see that there are 44 degrees, 83 degrees and 53 degrees in those angles.

And part E, use tracing paper to check that your triangles from parts B and D are congruent.

You'd do that by tracing over one of them and then when you turn that tracing paper around, either rotating it or reflecting it in some sort of way or a bit of both, the triangle should fit perfectly over the other constructed triangle.

And then question two.

Part A, I need you to construct a triangle that looks something like this.

Again, it might be in a slightly different orientation.

As an extra check, you could use a protractor to measure the angles, and because it's equilateral, it should all be 60 degrees.

Part B asks you to draw a perpendicular bisector through one the sides and it should look something like you can see on the screen here.

Then you need to think, why does your construction from Part B bisect an angle? Well, there's a few different ways you can explain this.

One is that because the other two sides are of equal length or you could think about how the two points where the needle of a pair of compasses were placed are equidistant from the angle.

'Cause that's what you do when you bisect an angle.

You first create two points that are equidistance away from where the angle is and then you continue your construction from there.

Part D, you need to use Pythagoras's theorem to calculate the shortest distance from a vertex to its opposite side.

The way you'll do that, by using these measurements here.

Your hypotenuse is eight centimetres, and the one on the shorter size is four centimetres.

So when you apply the Pythagoras's theorem, you'd get 6.

9 centimetres for the third remaining length.

And you can measure that with a ruler to see how accurate it is on your diagram.

In question three, you're gonna get an accurate drawing of a triangle that has a side length of six centimetres and is similar to the one you can see here.

Well, there are three possible solutions to this, depending on which side of the original triangle you choose to scale up to six centimetres.

The one on the left, that one has a scale factor of three because we've scaled up the two centimetres to make six centimetres.

The one in the middle has a scale factor of two because we have scaled up the three centimetre side to be six centimetres and the one on the right that has a scale factor of 1.

5 because we've scaled up the four centimetre side to be six centimetres.

Fantastic work today.

Now let's summarise what we've learned.

An accurate drawing of a triangle can be made when all three side lengths are known.

In other words, side, side, side.

The first side can be drawn accurately with a ruler and it doesn't necessarily matter which direction that side goes in unless you are asked to do it in a particular way.

A pair of compasses should be used to indicate the potential positions for the other sides by drawing arcs.

And what you're looking for is where those arcs intersect.

The arcs intersect at a point which the other two sides can be drawn to.

And all triangles with the same measurements are congruent.

Well done today.

Have a great day.