Lesson video

Hello and thank you for choosing this lesson.

My name is Dr.

Rowlandson and I'm excited to be helping you with your learn 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 angles and the side length between them.

And by the end of today's lesson we'll be able to do precisely that.

We'll be able to construct a triangle given two angles and a side length between them.

A previous keyword that will be useful during today's lesson is the word congruent.

If one shape can fit exactly on top of another using rotation, reflection, or translation, then the shapes are congruent.

The lesson will be broken into two learning cycles.

In the first learning cycle, we're going to be constructing triangles perfectly accurately by using dynamic geometry software and doing that we'll allow us to explore the theory behind the constructions.

In other words, knowing why those particular constructions produce a triangle in that particular way.

And then in the second learning cycle of the lesson, we're going to be using paper and a pencil and some other apparatus and think about how we can replicate the same processes again when using that equipment.

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 the information that is given to us, such as the size of two angles and the length of the side between them.

The constructions used in dynamic software can also be replicated by hand by using a pencil and some other apparatus such as a ruler and a protractor.

So let's get started.

We'll be using GeoGebra for our dynamic software in this lesson.

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

org and then find and press the Start Calculator button and click on the Graphing button to open up an options menu and click on Geometry.

Once you do that, it should produce a page that looks a bit like this where we've got a blank canvas on the right and some tools in a toolbar on the left, but there are not many tools to begin with when you first open this up.

So you can click on More to access more tools and you can see that more will appear and you can scroll down to the bottom to see the full range of tools available.

But if you want more tools still, there is another more button that you can click on again and that'll create even more tools for you if necessary.

So here we have a sketch of a triangle where one of the side lengths is eight units long and the angle at either side of it is 50 degrees and 42 degrees.

Or we can think about it as we have two angles, 50 degrees and 42 degrees, and the side length between them is eight units long.

And what we're going to do is use dynamic geometry software to make an accurate drawing of this triangle.

And let's start by drawing the first side.

To draw the first side, find the section of tools under the subheading of lines and click on Segment with given Length.

Click somewhere on the canvas to plot the first point and then type in the length of the side and press Enter.

So in this case we'll be typing in eight and that'll produce a line segment that is eight units long.

Now you can move either the points at either end of that line segment and turn the line segment around if you want to, but if you use that tool, the line segment will always remain exactly eight units long.

So we've got our first side.

Now let's draw the first angle.

Find a section of tools under the subheading of Measure and click on Angle with given Size, click on two points, but do note that the second point you click on will be where the angle is created.

So we click on the point on the right first and then click on the point on the left.

Then the angle appear on the left.

And then if you do that, it'll open up a box where you need to type in the size of the angle you want to draw, in this case 50 degrees.

And you can choose a direction which is either clockwise or anti-clockwise.

For anti-clockwise, on this programme it says counter-clockwise.

And then once you've done that press Okay, and it'll create a point and that point makes a 50 degree angle with the first two points that you drew.

However, that new point is not the third point on our triangle.

What we are not going to do now is join these up to create a triangle.

And that's because even though that point makes a 50 degree angle with the line segment in one direction, it does not necessarily make a 42 degree angle with the line segment in the other direction.

In other words, if we join up those points to make a triangle, one angle will be 50 degrees, but the other one will not necessarily be 42 degrees.

But what that new point does allow us to do is to draw the possibilities for the second side.

And to do that we're going to draw a ray line.

So find a section of tools under the subheading of lines, and click on "Ray." First, click on the point where the angle is and then click on the last point we just made.

And that'll draw a ray from our point of the angle through that point what we just made and off to the edge of the screen.

And that ray line shows the direction of the second side.

So now we're going to repeat the same process again in order to draw the second angle.

First we'll go to Measure and click on Angle with a given Size.

Then we'll click on each point on that first line segment that we made.

But this time click on it in the opposite order that you did last time.

We want, in this case here, the angle to be on the right hand side.

So we'll click on the point in the left first and then we'll click on the point in the right.

So the angle appears on the right hand side, it'll open up a box, we can type in 42 degrees.

And then this time we're gonna click on the opposite direction that we used last time.

So the first time we used anti-clockwise or counter-clockwise.

So this time we're gonna ask it to go in a clockwise direction and press Enter.

And that will create another point.

And our new point makes a 42 degree angle with the first line segment we drew.

So we can repeat the same process again for drawing the possibilities.

for the third side.

We can find a section of tools under Lines and click Ray.

Click on the point where the angle is, where the 42 degree angle is and then click on the new point that we made.

And that'll draw a ray line through that point.

And this ray line shows the direction of the third side.

So now we've got our first side drawn at eight units long and we've got the direction for the other two sides indicated by ray lines.

Can you see where a third point in this triangle is going to go? It's going to be at the point where those two rail lines intersect.

So let's now mark that point on with GeoGebra.

We're going to find the tools under Points and click on Intersect and then click on each ray line and that will show you with a point where the two intersect or it will create a point there for you to use later.

And then we can draw our triangle by going to the tools under Polygons and clicking Polygon.

And I click on each of the three points and that will create our triangle.

Let's check what we've learned.

Here we've got Laura who is using GeoGebra to accurately draw a triangle with angles 125 degrees and 32 degrees with the side length between them at five units.

And this is what she's drawn so far.

You can see that she's got the side length, which is five units long, and she's drawn the two angles and she's got ray lines.

And then Laura plots point C, which we can now see on the screen.

Which tool did Laura use to do that? Pause video while you're choose me the A, B, C, or D, and then press play when you're ready for an answer.

The answer is B.

She used the intersect tool to create that point with the two ray lines intersect.

So she's now completed her triangle.

What is the size of angle ACB? That is the unknown angle in this triangle.

Pause the video, why you work that out and press play when you're ready for an answer.

We can work out the size of this angle by subtracting the two angles that we know from 180 degrees or do 180 degrees subtract the sum of 125 and 32, and that'll give us 23 degrees.

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

This task consists of one question and here it is, you need to open up a web browser, go to GeoGebra.

org and open up a blank geometry page.

And then make an accurate drawing of each triangle that I represented by the sketches you can see on the screen here.

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

Okay, let's go through some answers.

For triangle ABC, it should look something a bit like this, however, yours might be in a different orientation, so as an extra check, what you could do is you could go to the tools that say Measure and click on the tool called Distance or Length.

And then you can click on each side of the triangle that has an unknown length and that'll tell you that side AC is 10.

7 units and side AB is 7.

1 units if you have drawn it accurately.

Triangle DEF should look something like this or in a different orientation and to double check that's correct you could work out the size of the two unknown side lengths using the same method.

And if you have drawn it correctly, it should be 10 units and 10 units.

This is an equilateral triangle.

And then triangle GHI should look something a bit like this or again in a different orientation.

To check these, you could work out the length of the two unknown sides, and if you've drawn it correctly, it should be 13.

7 units and 9.

4 units.

And then for triangle JKL, what makes this a little bit trickier is that you don't have the angles at either side of the line segment of a known length.

You're given two different angles, but what you could do is work out the third unknown angle by subtracting the two angles you do know from 180 degrees and that will give you 34 degrees.

And that means you can draw your triangle and it would look something a bit like this or in a different orientation.

To double check your answer, you can measure the length of the two unknown sides and there should both be nine units.

You're doing great so far.

Now let's move on to the second part of this lesson where we attempt to replicate what we've just done, but using pencil, paper and some other apparatus.

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

The triangle has two known angles and the side length between them is also known as well.

Here we have Lucas who wants to accurately draw the triangle on paper using a pencil and some other appropriate equipment.

He says, "I have a pencil, ruler and a protractor." How could Lucas draw the triangle accurately using this equipment? What steps could Lucas take? What would we do first and then what would we do? And then what would we do after that? Pause the video while you think about it and press play when you're ready to do this together.

Okay, I wonder what your thoughts.

Let's see how Lucas does this.

Lucas makes an accurate drawing of the triangle shown in the sketch.

He says, "I'll start by drawing the side with a known length." So he's going to use his rule of that.

He draws a side that is eight centimetres long.

He then says, "I'll use my protractor to measure a 50 degree angle and mark a faint point." So we have our protractor aligned at the point on the left of this line segment, we're gonna measure a 50 degree angle and that will be going anti-clockwise.

So we'll be starting at zero on the inside numbers go to anticlockwise to 50 and mark a faint point there.

And that point just shows where the 50 degree angle is.

So Lucas says, "I'll then draw a ray line through that point to show the direction that the second side will go in." Like so, and we've got our angle of 50 degrees.

He then says, "I'll use my protractor to measure a 42 degree angle at the other end of the line segment and mark it with a faint point." And if we look, the zero that we are using is on the outside scale of the protractor, which means we are going clockwise around that scale.

And 42 degrees would be about here.

So now what we've marked a faint point for where the angle is.

Lucas says, "I'll draw a ray line through that point to show the direction that the third side will go in." It looks something a bit like this and we have our angle of 42 degrees.

So what we can now see in our diagram is we've got the first side of our triangle drawn, which is eight centimetres, and we don't have the second and third side of this triangle fully drawn yet.

But what we do have is a faint ray line to show each of the directions.

Can you now see where a third point of our triangle will go? Lucas says, "I can now draw the triangle by using the intersection as its third point or its third vertex." That means our triangle would look something a bit like this.

We have a triangle with a 50 degree angle, a 42 degree angle, and the length of the side between them is eight centimetres.

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

Lucas says, "I could have drawn the two angles the other way around, so rather having the fifth degree angle on the left and the 42 degree angle on the right, we could have drawn it this way." Or he says, "I could have measured the angles below the side with a known length." So he would look something a bit like this or he could draw this triangle in any orientation he wants to and he would still be congruent.

So let's check what we've learned.

Here we have a diagram that shows part of a protractor and it's measuring an angle.

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

The answer is A, 25 degrees.

The zero that we are using in this angle is on the inside scale of the protractor.

So we are going anti-clockwise along the inside scale and that gives us 25 degrees.

So how about this angle now? What is the size of this angle according to the protractor? Pause video, while you choose and press play when you're ready for an answer.

The answer is A, 76 degrees.

The zero that we are using this time is on the outside scale and that means we'll be going clockwise along that scale to get 76 degrees.

Let's check how well you can do this yourself now.

You've got a sketch of a triangle ABC, where you've got an angle of 25 degrees, an angle of 76 degrees, and the length of the side between them is 10 centimetres.

Have a go at using a protractor and ruler to make an accurate drawing of this triangle.

Now it's not always easy to do first time, so don't worry if you find it a little bit tricky and just try your best of it and make it as accurate as you can.

Pause video while you do it and press play when you're ready to check the answer.

So here's what your answer could look like, however yours may be in a different orientation.

You can check your answer by measuring each measurement again.

So measure each of the angles using a protractor, measure the side lengths using a ruler or if you have a partner who can help you out, get them to measure each part again as well and see if it is accurately drawn.

And then looking at this triangle in a little bit more detail now.

What is the size of angle ABC, in other words, the unknown angle? Pause video while you work it out and press play when you are ready for an answer.

We can get our answer by subtracting the sum of the angles that we know from 180 degrees and that gives us 79 degrees.

Okay, so to now for task B, this task contains two questions and here is question one.

You have four triangles that are represented by sketches and what you need to do is take a sheet of plain paper, a pencil, a rule and a protractor, and make an accurate drawing of each triangle.

And then once you've done that for part B, I'd like you to go back to each those four triangles and then look at the side which has a known length.

For each one, construct a perpendicular bisector for that particular side.

And then look at for which triangles does the perpendicular bisector intersect an angle? And why does it do this for those particular triangles? What you might want to do while doing this is each time you are about to draw a perpendicular bisector, anticipate whether or not you think it would intersect the angle which is opposite and consider why.

And then when you constructed it, you can see if you are right or not.

Either way, pause the video, while you this and press play when you're ready for question two.

And here is question two.

You have a sketch of triangle ABC, and what you need to do first for part A is to make an accurate drawing of that triangle.

And then once you've done that for part B, you need to calculate the lengths of side AB and side CA, if you get any non indigent answers round to one decimal place.

And then measure each side with a ruler to check the accuracy of your drawing.

Now you might be wondering at how you could possibly calculate the lengths of AB and CA.

What might help is if you work out the third remaining angle first, and that might give you some clues to what sorts of aspects of maths you might use to work out those missing sides.

Pause video while you do that and press play when you are ready to go through some answers.

Okay, let's take a look at some answers.

So for question one, triangle ABC should look something a bit like this.

Again, it can be in different orientations, so you can check it by measuring the two unknown lengths with a ruler and you should have 10.

7 centimetres and 7.

1 centimetres.

And once you've drawn your perpendicular bisector, you'll notice that it does not intersect an angle.

And then for triangle DEF, once you've drawn it, it should look something a bit like this for an extra check you can use your ruler to measure the two unknown side lengths and they should both be 10 centimetres.

It's an equilateral triangle.

And then when you draw your perpendicular bisector, it should look something a bit like this and this perpendicular bisector does intersect an angle.

For triangle GHI, it should look something a bit like this or in a different orientation, for your extra checks, you can measure the other two unknown sides which are 13.

7 centimetres and 9.

4 centimetres.

And when you draw your perpendicular bisector, it should look like this and it does not intersect an angle.

And then for triangle JKL, you need to work out the third remaining angle first, and then you can draw it and it should look something a bit like this or in any other orientation.

And as an extra check, you can use your ruler to measure the length of the two unknown sides and should both be nine centimetres.

And when you construct your perpendicular bisector, you'll notice that it does intersect an angle.

So you saw some cases where the perpendicular bisector did intersect an angle and some where it didn't.

When did it and when did it not and why? Well, the perpendicular bisector intersects angle in triangle DEF and triangle JKL, and here are some examples of explanations for why.

You could say from the angles of the vertices at either end of the bisected side in these triangles are equal, the triangle DEF, both of the angles were 60 degrees.

For triangle JKL, both of those angles were 34 degrees.

And another explanation could be that these triangles are either isosceles or equilateral, therefore they are symmetrical with the perpendicular bisector as the line of symmetry.

Then in question two, you have to make an accurate drawing of triangle ABC.

It should look something a bit like this or in a different orientation, but this time it was up to you to check this yourself.

You need to work out the length of side AB and side CA.

The way to do that is if you first work out the missing angle, you'll notice it's 90 degrees, which means this is a right angle triangle.

If it's a right angle triangle, and we have a side length and some angles, it means we can use trigonometry in order to work out the length of the missing sides.

One way you can show this in your working could be to let the length AB equal X and the length CA to equal Y or you can use different letters.

And then to work out the value of X, you could use sin ratio, which would be sin of 30 degrees equals x divided by 12, and then rearrange that equation, you get X equals six, so the length of AB is six centimetres.

And then to work out the value of Y, you could use the co-sin ratio.

You could do cos of 30 degrees equals Y over 12 and then rearrange that and you would get Y equals 10.

4 when it's round to one decimal place.

Therefore the length will be 10.

4 centimetres.

Now in this method we have chosen to use the 30 degree angle each time and trigonometry to work out the values of X and Y.

But there are alternative methods.

You could use trigonometry with the 60 degree angle.

For X, that would be co-sin is 60 degrees equals X over 12.

And for Y it would be sin of six degrees equals Y over 12, or you could work out one of the lengths and then use Pythagoras theorem to work out the third length if you want to instead.

However you choose to do it, you should still get the same answers 6 and 10.

4 and then you can measure those on your diagram and see how accurate they are.

Fantastic work today.

Now let's summarise what we've learned.

An accurate drawing of a triangle can be made when the size of two angles are known and the length of the side between them is known.

In the configuration angle-side-angle.

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

And then both the angles should be measured accurately with a protractor.

And then by drawing ray lines through each angle marker that you make, you can see where they intersect.

And this is the point where the two remaining sides should be drawn to.

And don't forget that all triangles that have the same measurements are congruent.

Well done today.

Have a great day.