Lesson video

Hello, and thank you for choosing this lesson.

My name is Dr.

Rowlandson, I'm excited to be helping you with your learning today.

Let's get started.

Welcome to today's lesson from the unit of loci and constructions.

This lesson is called constructing a triangle given two side lengths and the angle between them using compasses.

And by the end of today's lesson, we'll be able to construct a triangle given two of the required side lengths and the size of the angle between them.

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 either these words mean and then press play when you're ready to continue.

The lesson is broken into two learning cycles.

In the first learning cycle, we're going to be using dynamic geometry software in order to draw triangles perfectly accurately, and that will allow us to explore the theory behind the constructions while we're doing that.

In other words, understand what it is we are doing when we construct these triangles and why these particular constructions produce the end result.

Then the second part of the lesson, we're going to apply what we've learned by a think about how to replicate the same processes again when drawing triangles on paper with a pencil and how using particular apparatus allows us to replicate those particular constructions.

Apparatus such as a pair of compasses and a protractor.

But let's start off with drawing triangles with dynamic geometry software.

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 you may be given two side lengths and an angle.

The constructions used in dynamic software can be replicated by hand using a pencil and apparatus such as a ruler, a pair of compasses, and a protractor.

And as we go through the first part of this lesson, what you may wanna think about in the back of your mind is what equipment would you use to replicate the same process on paper? When would you use a protractor? When would you use a pair of compasses and so on.

So let's get started.

The dynamic geometry software that we're going to use during today's lesson is GeoGebra.

So open up a web browser and go to geogebra.

org and then find and press the start calculator button.

When you do that, find and press the Graphing button to open up the options menu and click on Geometry.

That'll open up a page that looks a little bit like this where you've got a blank canvas on the right and you've got some tools in a toolbar on the left of the screen.

There's not many tools to begin with though, so what you can do is click on more to access more tools and you can scroll down to see what tools are available.

And if you need more tools than that, you can click on more again to access even more tools.

So let's now talk about how we can use dynamic geometry software in order to make an accurate drawing of this triangle represented in the sketch on the screen.

It has one side length that is 7 units long, another side length that is 8 units long, and the angle between those two side lengths is 50 degrees.

We're not told the length for the third side and we're not told the other two angles, but we can still construct that triangle perfectly accurately using that information.

Let's start by drawing the first side, the one which is 8 units long.

To do that, find the section of tools under the subheading of lines and click on segment with given length.

Then click somewhere on the canvas to plot your first point and type in the length of the side and press enter.

So, in this case, it's gonna be 8 and that'll produce a line segment that looks something like this, which is 8 units long.

Now, if you move out of the points on either side of that line segment, that's absolutely fine and the line segment will remain 8 units long.

So let's now draw the angle of 50 degrees.

To do that, find the section of tools under the subheading of measure and click on angle with given size.

Now, the angle will be between three points.

You need to click on two of the points and GeoGebra will create the third point for you and it'll be at the angle that you choose it to be.

So the next thing you'd do is click on two points and note that the second point you click on will be where the angle appears.

So if you click on the right-hand point first and then the left-hand point, the angle will appear in the left corner.

So, next thing to do is type in the angle you want.

In this case, we want 50 degrees.

I can choose either clockwise or counterclockwise or anti-clockwise in other words, and then press okay.

And if you do that, it'll plot a point like we can see on the screen here now where the angle between those three points is 50 degrees.

So now let's draw the second side.

Firstly, let's draw a faint construction line that goes from the angle through the new point we created.

To do that, find the section of tools under the subheading of lines and click ray.

And then, first, click on the point where the angle is, so in this case, the bottom left corner, and then click on the last point you made and that'll draw a ray line that looks a bit like this.

So the next thing we want to do is mark a point somewhere on that ray line that is exactly 7 units away from the points where the angle is.

To do that, we're going to draw a circle.

Find the section of tools under the subheading of circles and click circle: centre and radius.

Click on the point where the angle is because we want it to be 7 units away from that angle and type in the length of the side, which in this case is 7, and press enter.

It draws a circle a bit like this.

We can't see the whole circle on the screen, but that's absolutely fine.

We can see the important part of the arc and that is where it intersects the ray line.

Now, every point along the arc is exactly 7 units away from the point where the angle is, but only one of those points makes a 50-degree angle with the first line segment we drew, and that is the point where the two of them intersect.

So to map that point on our diagram, we're going to find the section of tools under the subheading of point and click intersect.

Click on the circle and click on the ray line and that'll show the point where the two intersect, and that point is exactly 7 units away from the point where the 50-degree angle is.

So finally, we've got three points for our triangle.

The first two we made with the first line segment and the last point we just made, we need to complete our triangle.

Find the section of tools under the subheading of polygons and click polygon, and then click on the three points to make your triangle.

And that'll produce something that looks something like this.

So let's check what we've learned.

Jun is using GeoGebra to accurately draw a triangle with side lengths 5 units, 4 units, and an angle of size 100 degrees between them.

Here's what Jun has drawn so far.

He's got a line segment from A to B, which is 5 units long, and he's got an angle which is 100 degrees.

Which tool did Jun use to construct the angle? Pause while you choose from A, B, C, or D, and press play when you're ready for an answer.

The answer is A.

He used the tool which is called angle with given size.

Now, when Jun constructed the angle using the angle with given size tool, which point did Jun click on first and which point did he click on second? Pause video while you write that down, and press play when you are ready for an answer.

Well, you need to remember that the second point you click on is the point where the angle appears, so you clicked on the point B first and then clicked on point A second.

True or false? If Jun connects the three points, what we can see on the screen here, the triangle will have lengths 5 units and 4 units.

Is that true or is it false, and choose a justification.

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

The answer is false and the reason why is that the top point is not necessarily 4 units away from point A.

GeoGebra will have created A point that makes 100 degrees with the other two points, but it will not necessarily be the distance away that you want it to be.

So, bear that in mind.

Jun draws a ray line and a circle with radius 4 units and then marks the intersection with point C, which line segment would have a length of 4 units? Choose from either A, B, or C, pause while you do it, and press play for an answer.

The answer is B, the line segment AC would have length 4 units.

That is the radius of the circle.

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

This task comprises two questions, and here is question one.

You need to follow the instructions in this question to produce two triangles and then explain why those two triangles are congruent.

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

And here is question two.

Once again, go onto GeoGebra, open the blank geometry page, and this time, you are going to make accurate drawings of these three triangles that are represented by sketches.

Pause video while you do that, and press play when you are ready to look at some answers.

Okay, let's see how we got on.

With question one, for parts A, B, and C, that should lead you to produce a triangle that looks something a bit like this.

You'll have one length which is 8 units, one length which is 10 units, and an angle between them that is 130 degrees.

Now, your triangle may be in a different orientation to this one.

That's absolutely fine.

You can turn yours around to see if it looks like this one or you can do an extra check.

To do an extra check, you could go on the toolbar, find the tools under measure, and then click on the tool that says distance or length and then click on the third side of your triangle which has an unknown length.

Now, if you've plotted your triangle accurately, that length should be 16.

3 units.

And then parts D and E lead you to create a triangle that looks something a bit like this.

Again, yours may be in a different orientation, that's absolutely fine.

To check it, you can go on tools, measure, click on the distance or length tool and then click on the third side and that again should be 16.

3 units.

And then part F, you need to explain why these two triangles are congruent.

Well, the two side lengths and the angle between them are the same in each triangle.

In other words, you've got a side, angle, and side situation there which shows that they are congruent.

Now, it's worth to bear in mind that you may have used some tools to find other measurements such as find the length of the third side or find the size of your angles.

If you have done that, it does open you up to use alternative justifications for why they're congruent, such as for example, you could say all three sides are the same length, but with the information you're given, you can say they're congruent based on the side, angle, side.

Then question two, when you create a triangle that is an accurate drawing of the one in part A, it should look something like this or in a different orientation.

When you do your extra check by checking the length of the third side, it should be 11.

7 units.

With part B, you're not given the size of the angle that is between the two sides with known lengths, but what you could do is subtract the two angles that you are given from 180 degrees to find that it's 30 degrees between the two side lengths that you know and that means you can then produce this triangle or one that is in a different orientation to that.

If you want to double-check it's correct, you can find the length of the third side and it should be 7.

6 units.

Then part C, once again, you don't know the size of the angle which is between the two sides with known lengths, but what you can see is that this triangle is isosceles, two of the sides have the same length, which means two of the angles have the same length.

Therefore, you could work out the size of the angle that's between the two sides that you know and that would be 40 degrees, and then you can produce a triangle you can see on the screen here or one in a different orientation.

And if you want to do an extra check and find the length of the unknown side, it should be 8.

2 units.

Well done so far.

Now let's move on to the second part of this lesson where we're going to consider how we can replicate the same processes again when using pencil and paper and some other apparatus.

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

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

Now, Sam says, "I have a pencil, ruler, protractor, and a pair of compasses." How could Sam draw the triangle accurately using this equipment? Think about what equipment Sam might use to draw the first line segment or to draw the angle or to draw the arc you see.

What steps would Sam take when doing this? Pause video while you think about this, and press play when you're ready to do it together.

Let's take a look at this together then.

The sketch shows the lengths of two sides and one angle of a triangle and they are arranged into side, angle, side.

Sam makes an accurate drawing of this triangle.

Let's see what Sam does.

"First, I'll start by drawing one of the sides with a known length." So we can either draw the side which is 8 centimetres long or 7 centimetres long.

Sam chooses to draw the one which is 8 centimetres long.

So we can start by plotting the point for R, then use a ruler to measure 8 centimetres and plot the point for Q.

We've now got one side.

Sam says, "I'll use my protractor to measure 50 degrees and mark a faint point, right here." Then Sam says, "I'll draw a ray line through that point to show the direction that the second side will go in," like this, and now we've made an angle that is 50 degrees.

Then that line segment we can see there that goes from R and goes diagonally upwards, if that's a ray line, that'll go on for infinity or off to the edge of your page, it's not 7 centimetres long.

That's the key think.

So what we now need to do is figure out where on that line is 7 centimetres away from point R.

So Sam says, "I could open up a pair of compasses to 7 centimetres apart and draw an arc around point R, like this." Every point along that arc is 7 centimetres from point R, but only one of those points also makes an angle of 50 degrees with RQ.

That's the point where they intersect.

Sam says, "The point where the arc intersects the ray line will be the third point of my triangle." So that'll be point P.

"And finally, I'll connect points P and Q with a line segment to complete my triangle." So now we have a triangle with two side lengths, 8 centimetres and 7 centimetres, with the angle between them as 50 degrees and we've chosen to draw it in this particular orientation.

But other triangles with the same three measurements in the same configuration, side, angle, side would be congruent to this one.

So Sam says, "I could also drawn this triangle in other orientations." For example, after we've drew the first side, which was 8 centimetres, we chose to put the 50-degree angle on the left, but we could have put the 50-degree angle on the right instead and it would look something a bit like this or we could have put the angles below that side so it would look something like this or like this, or we could have chosen to draw the 7-centimeter side horizontally instead.

Sam says, "All the possible triangles with these measurements are congruent." Now, Sam's method replicated the process for how the triangle could be constructed using dynamic geometry software, and Sam's construction lines would look pretty similar to the ones we saw on GeoGebra.

But is there anything that Sam could have done differently? Pause video while you think about it, and press play when you're ready to continue.

Sam says, "When I got to this stage where we've drawn the first side and we've marked the point where the angle would be 50 degrees and we've taken our protractor away, rather than drawing a circle or arc with a pair of compasses, an alternative method could have been to align my ruler with R and the faint point and then measure 7 centimetres, like this." And then, we could draw the third side from there.

Let's check what we've learned.

Here we've got a diagram that shows part of a protractor, measuring an angle.

What is the size of the angle? Pause while you choose, and press play when you're ready for an answer.

The answer is A, 63 degrees.

Here is a different angle.

What is the size of the angle according to a protractor? Pause while you choose and press play when you're ready for an answer.

The answer is C, 126 degrees.

And here we have Alex who is making an accurate drawing of triangle PQR, which is shown in the sketch on the top right of the screen.

He's drawn one side then already, 8 centimetres, and he says, "I've marked a point next to 55 degrees on my protractor.

I then connected that point to R and Q to complete my triangle." Why is this wrong? Pause video while you write down a sentence for why this is wrong, and press play when you're ready for an answer.

This is wrong because point P is not necessarily 7 centimetres away from point R.

Depends on how big his protractor is.

Let's now construct another triangle.

Here we have Izzy who is making an accurate drawing of the triangle represented in the sketch.

We've got one side length which is 8 centimetres, another which is 6 centimetres, and the angle between them is 60 degrees and the fact that the angle is 60 degrees is important for what we are about to do because it opens up some alternative methods.

Izzy says, "I think that I could do this without using a protractor.

But I would need a ruler and a pair of compasses." Can you think about how Izzy might construct a 60-degree angle without using a protractor, but using a pair of compasses and a ruler? Perhaps pause the video while you consider what we're about to do and then press play when you're ready to continue together.

Let's see what Izzy does.

She says, "I could start by using a ruler and a pair of compasses to faintly draw an equilateral triangle with side lengths 8 centimetres because each angle in an equilateral triangle is 60 degrees.

So we've got one side length drawn here, which is 8 centimetres.

We could get a pair of compasses and open it so that the needle is at one end of the line segment and the pencil tip is at the over end.

Draw an arc, move our pair of compasses so that the needle is on the other end of the line segment, draw another arc, and then the point where those two arcs intersect will be exactly 8 centimetres from either side of that line segment and that will create an equilateral triangle, a bit like this.

And because it's equilateral, it means all the angles are the same size, so each angle would be 60 degrees.

So we have constructed a 60-degree angle here without using a protractor, but we haven't got the triangle we want because the triangle we want does not have sides that are all the same length like the triangle we've constructed.

So what's Izzy going to do next? She says, "I'll then measure 6 centimetres along the side on the left to find the top vertex of my triangle." We can see with Izzy's construction, that two of the sides of the triangle that Izzy has drawn so far are quite faint, and that's because those two are not the final sides of her triangle.

So she's gonna measure 6 centimetres along one of them and make that a side for her triangle and then draw the third side from there.

And now we've got our triangle with 8 centimetres and 6 centimetres with 60-degree angle in between them without using a protractor.

So let's review what Izzy did then.

She drew an equilateral triangle and then she adapted the triangle to draw the triangle that is shown in the sketch.

But are there any steps in Izzy's method that could be done differently? Still using broadly the same method using a pair of compasses and not a protractor, but is there anything that Izzy could have done a bit quicker or were there any steps that Izzy did that were unnecessary? Pause video while you think about this, and press play when you're ready to continue together.

Izzy says, "I didn't need to draw a complete equilateral triangle," like she did here.

She says, "I could have just drawn a ray line from one point through the intersection, measure 6 centimetres along that, and then you've got your triangle." In other words, she didn't need to draw the third side of the equilateral triangle.

So let's check what we've learned.

Here we have a pair compasses that is used to draw an arc.

What is the value of X that is labelled on that diagram now? Pause video while you write that down, and press play when you're ready for an answer.

The answer is 10.

That length is 10 centimetres.

Now, a pair of compasses is used to draw another arc.

What is the value of theta? In other words, what's the size of that angle that has just been made? Pause video while you write it down, and press play when you're ready for an answer.

The answer is 60.

The angle is 60 degrees.

You can see that those two arcs are the same size and the radius is the same size as a line segment that is 10 centimetres.

Therefore, if you join them up, you create an equilateral triangle and that means the angle would be 60 degrees.

Here we have Jacob who wants to draw a triangle with side lengths 10 centimetres and 13 centimetres, with the angle between them at 60 degrees.

The diagram shows what he's done so far.

He's drawn the first side at 10 centimetres.

He's constructed a 60-degree angle, and you can see that there are three points currently marked on that ray line labelled A, B, and C.

Which of those points should he use to draw the two remaining sides of the triangle? Pause video while you choose and press play when you're ready for an answer.

The answer is point A.

You can see on the ruler that point is 13 centimetres away from the point where the angle is.

But even if you didn't have the ruler on the screen, you could deduce that out of those three points, only point A could be 13 centimetres away because point B is the point where the two arc intersect and that is 10 centimetres away.

Therefore, A is the only point which is greater than 10 centimetres from the point where the angle is.

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

This task contains three questions, and here is question one.

Take a sheet of paper, a pencil, a ruler, and a protractor and follow the instructions in these parts of the questions to construct two triangles and then use a piece of tracing paper to check whether your triangles are congruent.

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

And here is question two.

Take a sheet of plain paper, a pencil, ruler, and this time, a pair of compasses, not a protractor.

And using those, make an accurate drawing of the triangle represented in this sketch.

And then once you've done that, if you have previously learned about the cosine rule in non-right-angled trigonometry, use that to calculate the length of the third side and then measure the length of the third side on your diagram with a ruler to check how accurately it is drawn.

If you haven't previously learned about the cosine rule, don't worry about it.

Just miss this part out.

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

And here is question three.

Once again, take a plain piece of paper, a pencil, and this time, you can choose the other equipment that you use.

You've got a sketch of a kite where the length of DB, the diagonal length is 9 centimetres, and you can see some other measurements shown on the sketch as well.

Make an accurate drawing of this kite based on what you've learned in today's lesson.

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

Okay, let's go through some answers now.

With question one, after you constructed your first triangle, it should look something like this or it may be in a different orientation.

And if you want to double-check, you can measure the length of the third side of triangle and it should be 16.

3 centimetres.

Also, if you completed the first part of this lesson with task A, then this triangle should look like one of the triangles you constructed in that lesson.

You can always compare them if you have them both at hand.

In parts D and E, that should lead you to draw a triangle that looks something a bit like this or in a different orientation.

As an extra check, the third length should be 16.

3 centimetres.

And then, when you use a piece of tracing paper to check that your triangles from parts C and E are congruent, what you should find is that the traced angle should fit perfectly over both constructed triangles by using reflection or rotation or a bit of both.

In question two, first you need to make an accurate drawing of this triangle and it should look something a bit like this or in a different orientation.

And then if you use the cosine rule to work out the length of the third side, you should get 8.

9 to 1 decimal place for 8.

9 centimetres.

And then if you measure it, hopefully, it'll be 8.

9 centimetres on your diagram as well.

And then question three, you have to make an accurate drawing of this kite.

Well, you could break this up into two congruent triangles by drawing a line segment from D to B.

So first, you could construct one of those triangles, triangle ABD where they got the lines of length, 9 centimetres and 5 centimetres, and the angle between them have been 30 degrees because it's half of the angle shown in the kite.

And once you've drawn that triangle, you could draw another one that is congruent to it underneath.

So now you've got your kite DABC.

As an extra check, what you could do is measure the length of AB and measure the length of BC, they should both be 5.

3 centimetres.

Fantastic work today.

Now let's summarise what we've learned.

An accurate drawing of a triangle can be made when the lengths of two sides are known, as well as the angle in between them, in the configuration side, angle, side.

The first side can be drawn accurately with a ruler, and it doesn't matter which direction that first side goes in.

Then, a pair of compasses can be used to indicate potential positions for the second side with a known length.

And a protractor can be used to measure accurately the desired angle.

And once you've done that, you'll have all the pieces you need to construct your triangle.

But also worth to bear in mind that all triangles with the same measurements are congruent.

Well done today.

Have a great day.