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 constructing a triangle given two side lengths and an angle not between them.

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

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 they mean, and press play to continue.

The lesson is broken into two learning cycles.

In the first learning cycle, we're going to be using dynamic geometry software to draw triangles with perfect precision, and this is so that we can explore the theory behind the constructions and understand why they work in producing the triangles we want.

And then the second part of the lesson, we're going to think about how to replicate those processes by hand using a pencil, paper and some other apparatus.

Let's start off with drawing triangles with dynamic 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 the length of two sides and the size of an angle that is not between them.

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

The dynamic software that we'll be using will be GeoGebra.

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

org.

And then find and press the Start Calculator button.

Find and press the Graphing button to open up an options menu, and choose the Geometry section.

That'll create a page that looks something a bit like this.

You've got a blank canvas on the right and some tools on the left, but there aren't many tools there.

So you can click on More to access more tools, and that would show more tools, which you can scroll down and see what's available.

If you still don't have the tool you want available there, you can click on More again to access even more tools if necessary.

Now, the steps that we use to construct a triangle can differ depending on what information is available to us.

For example, here, we have four different triangles where each time the information that's given to us is different.

In one example, we're told the length of each of the three sides.

In another example, we're told the length of two of the sides and the angle that's between them.

In the third example, we are told the two angles and the length of the side that's between them.

And in the fourth example, you're told that it's a right angle triangle, you're told the length of the hypotonus, and one of the other sides.

And based on these pieces of information, we would construct our triangle differently using slightly different order of steps.

Now, if you want to, you could memorise the step-by-step procedure for how to construct each of these different types of triangles, but that could be a little bit overwhelming.

There's a lot of steps to learn there.

Also, you'd then be limited to what you could draw.

You could only really draw triangles where they are giving you the types of information they've given you in these particular configurations here.

What happens if you are given a triangle where the information provided is not one of these four configurations? So rather than memorising entire methods for constructing each type of triangle, it can be helpful to consider what the individual steps of a construction achieve.

That way, you can think about taking different steps from different methods and combining them in such a way so that you can deduce new methods for yourself.

Let's take a look at a few really key tools that we often use when constructing triangles using GeoGebra.

One tool is the Segment with Given Length tool.

We can draw the first side of a triangle using this tool, and it can be useful when the direction of the side doesn't matter.

Another tool is the Circle with Centre and Radius tool, that is you click where the centre goes and you type in the radius you want for your circle.

We can consider all the possible positions for a side by drawing a circle with this tool.

And it can be useful particularly for drawing a second or third side of a triangle with a given length.

And the third tool is the Angle with Given Size tool.

We can draw an angle by using this tool, and then draw ray lines.

And this shows what directions the sides on either side of an angle will take.

So, here we have a sketch of a triangle where we know two lengths, six units and eight units, and we know the size of an angle, but it's not between those two units, it's 40 degrees.

Alex is planning to make an accurate drawing of his triangle on GeoGebra.

So what steps could Alex take? What might Alex do first, and then what might he do, and then what else might he do? Pause while you think about that, and press play when you're ready to continue.

Let's see what Alex is planning to do.

One possible plan could be to first draw the first side, and that's using this Segment with Given Length tool.

And then he could show all the possible positions that the second side might go by drawing a circle using the Circle, Centre and Radius tool.

And then he could show the direction that the third side will go by drawing the angle, and that'll be using the Angle with Given Size tool.

And then he can look for where the construction lines intersect, and that would help him to complete his triangle.

Alex says, "I could also do steps two and three in the opposite order." So he could draw the angle first and use a ray line to show what direction the unknown length will take.

And then at the other end of the line segment, he could draw a circle to show the different positions which these six unit length could take.

So let's work through this together now.

To draw the first side, find a section of tools under the subheading lines and click Segment with Given Length, and then click somewhere on the canvas to plot your first point and type in the length other side you wish to draw and press Enter.

Let's do eight.

And it draws a line segment that is eight units long.

Now to find the possibilities for the second side, let's find the section of tools under the subheading circles and click Circle, Centre and Radius.

Click on one of the points you've already plotted.

For example, the point on the right hand side and type in the length for the second side and press Enter.

So in this case, we'll type in six.

You draw a circle, it looks a bit like this.

Now we can't see all the circle on the screen here, but we can see the important part of it.

Every point along this arc is exactly six units away from the point on the right end of that first side.

Then to draw the angle, find the section of tools under the subheading Measure and click Angle with Given Size.

Click on each point on the first side that we drew, but note that the angle will be at the second point that you click on.

So if we want the angle to be on the left hand side of that first side, we wanna click on the point on the right first and then the point on the left.

And then we need to type in the size of the angle and press Enter.

In this case, 40 degrees.

And it creates a third point there.

That third point is not going to be the third point of our triangle.

That just shows where the angle of 40 degrees would go through.

Let's now draw a ray line through that point to show the possibilities for the third side.

Find a section of tools under the subheading Lines and click Ray.

Click on the point where the angle is and then click on the last point you made.

And that draws a ray line through that point.

And that ray line shows a direction that the side with the unknown length will go in.

Now let's review what these construction lines show us about the triangle.

The ray line shows us the direction that the side with the unknown length will take.

We don't know how long that side will be, but it will go in that particular direction in relation to the first side we drew.

The arc shows us every single point that is exactly six units away from the right hand side of the first side we drew.

So what we are looking for is where the possibilities for the second and third side are both true, and that is the points where the arc and the ray line intersect.

So find a section of tools under the subheading of Point and click Intersect.

Click on the circle and the ray line to show the points where they intersect.

This provides us with two points.

These two points are both exactly six units away from the right hand side of that first side we drew.

And both of these two points would make an angle of 40 degrees with the first side we drew as well.

That means, to complete the triangle, find a section of tools under subheading Polygons and click Polygon, and click on the three points to make your triangle.

Alex says, "But there are two points with a ray line intersects a circle.

Which one should I use?" Well, you could use either.

Using either intersection will construct a triangle that satisfies the measurements given in the sketch.

In both of these cases, you can see on the screen here, the side at the bottom, which we can see, is eight units long.

That's the first one we drew.

The angle is 40 degrees because that's the angle between the first side and the ray line.

And the side on the right of the triangle is six units because that's the distance between the centre of the circle and its circumference.

So both of these triangles, they look different, but they do satisfy the measurements given.

Alex says.

"The triangle on the right looks more like the sketch, but they both have the measurements given in the sketch." So unless you are told to make it look in a particular way, either of these will do it.

So that means when we construct a triangle given two side lengths and the size of an angle that is not between them, it may produce two possible triangles that satisfy the measurements.

Not always two, but sometimes two.

And it's worth the spirit in mind that these triangles are not congruent.

They are incongruent because the unknown measurements are not the same in each triangle.

We can see, but for the triangle on the left, the unknown length is very small compared to the unknown length for the triangle on the right.

Even though we don't know what they are, can definitely see that it's smaller and you can also see the angles are different as well.

For the triangle on the left we can see it has an obtuse angle, that's the angle at the top.

But for the triangle on the right that doesn't contain any obtuse angles.

So they are not congruent, they are incongruent.

If you'd like to explore this more and see in which situations you get two possible triangles and in which situations you only get one triangle, you could click on the link on the slide deck that takes you to this GeoGebra file and allows you to move things around and play around with it to see what you can deduce from it.

So let's check what we've learned.

You've got three tools from GeoGebra.

Match the tools with what they're useful for when constructing a triangle.

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

Let's see how we got on.

When drawing the first side of a triangle, where the direction of the line doesn't matter, we can use the Segment with Given Length tool.

When considering all the possible positions for a side when drawing a second or third side at a given length, we can use the Circle: Centre and Radius tool.

And when drawing an angle to show what direction the sides on either side of an angle will take, we can use the Angle with Given Size tool.

True or false? In this diagram, triangle ABC and triangle ABD are congruent triangles.

Decide whether it's true or false and choose a justification.

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

It's false and that's because it is possible to construct two incongruent triangles with a side length of 12 units and 15 units with a 50 degree angle opposite the 12 unit side.

Here you've got a diagram with a construction of a triangle, which side has a length of five units? Pause while you choose and press play when you're ready for an answer.

The answer is a, which is side AB.

Which side of the triangle is also the radius of the circle? Pause while you choose and press play for an answer.

The answer is b, which is side BC.

And finally, which angle is 50 degrees? Pause while you choose and press play for an answer.

The answer is b, which is angle BAC.

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

This task contains 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.

I need you to make an accurate drawing of each triangle, based on the measurements given in each question.

And those are questions a, b, c, and d.

Now, in some questions it is possible to construct two incongruent triangles.

So in these cases, draw both triangles.

In some questions, it's only possible to draw one of them.

That's fine, just draw that one.

However, in some questions, it is impossible to construct any triangles.

So in these questions, draw the construction lines and then explain why the triangle is impossible.

Pause the video wile you do this and press play when you are ready for answers.

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

For a, you can construct two different triangles given those measurements and this is what they would look like.

Your answers might be in a different orientation and that's absolutely fine.

We still have two triangles that are incongruent to each other.

For b, you can only construct one incongruent triangle and it would look something a bit like this or in a different orientation.

For c, you can only construct one incongruent triangle again and it would look something a bit like this or in a different orientation.

And then for d, hmm, for d you cannot construct a triangle given those measurements that are provided to you.

And as for why, it could be explained in a few different ways.

Here are some explanations.

One is that there is no point when the circle and the ray line intersect.

Another is that as the angle is obtuse, then side KL would need to be longer than side JK in order for the triangle to be possible.

That would make the circle big enough so that it does intersect the ray line at least somewhere.

Or if you've previously learned about the sine rule for non-right angle trigonometry, you could use that.

You could set up an equation using the sine rule and then show that it has no solutions, and that would show that the triangle is impossible as well.

Well done so far.

Now let's move on to the second part of this lesson.

Where we attempt to replicate these same processes again but by hand and using other apparatus.

When drawing a triangle with pencil and paper, the following three tools can be particularly helpful during its construction.

One is a ruler.

We can draw the first side of a triangle by measuring it with a ruler and this can be useful when the direction of the side doesn't matter.

Another tool is a pair of compasses.

We can consider all the possible positions for a side by drawing a circle or an arc with a pair of compasses.

And this can be useful for drawing a second or third side at a given length.

And a third tool is a protractor.

We can draw an angle by measuring it with a protractor and then drawing faint ray lines with a ruler.

And this shows what directions the sides and either side of the angle will take.

So here we have a sketch of a triangle where one side is six centimetres, another side is eight centimetres, and an angle that is not between them is 40 degrees.

Alex is planning to make an accurate drawing on paper of the triangle represented in this sketch.

And his plan is below.

His plann is to draw the first side and then show the possible positions for the second side by drawing a circle or an arc.

And then he's going to show the direction of the third side by drawing the angle and then he is gonna look for where the construction lines intercept.

What equipment would he use for each of the first three steps? Pause video while you think about this and press play when you're ready to continue.

Well for the first step when he draws the first side, he could do that using a ruler.

For the second step, when he wants to show the possible positions for the second side by drawing an arc or a circle, he could do that using a pair of compasses.

And for the third step, when he wants to show the direction of the third side by drawing the angle, he could do that with a protractor.

Now Alex says, "I could also choose to draw the angle before the arc." He could swap steps two and three around.

He says, "This could make it easier to judge where to draw my arc." Once he's measured his angle and drawn a faint ray line to show the direction that the unknown side would take, then he can make his arc much smaller in the next step.

So, he doesn't have to draw a full circle, doesn't have to draw a very big arc, he can just draw a very small arc which intersects that particular ray line.

You'll see what I mean when we get to that point.

So Alex makes an accurate drawing for the triangle in the sketch.

He says, "I'll start by drawing one of the sides with a known length." He uses his ruler to draw a side which is eight centimetres, like this.

He says, "I'll use my protractor to measure 40 degrees and mark a faint point." Like this.

He then says, "I'll then draw a ray line through that point to show the direction that the second side will go in." Like this.

So we've got the first side drawn.

We've got a faint line showing possibilities for the unknown length side.

We still need to draw the six centimetre side.

He says.

"I could open up a pair of compasses to six centimetres apart and draw an arc around the point at the other end of the side" Like this.

Now we can see that the arc intersects the ray line, so his arc only needs to start a little bit before the ray line and it only needs to end a little bit after the ray line.

If he hadn't drawn that ray line already, if he was drawing the arc first, he wouldn't really know quite how big to make it.

He'd probably need to draw the full circle or at least a semicircle with it.

So that's why it can be helpful to draw the angle first, but it doesn't really matter either way.

What we can see is that there are two points where the ray and arc intersect.

Either of these could be used to create a triangle of the measurement shown in the sketch.

One triangle could look something a bit like this.

And this is the triangle that looks the most like the one in the sketch, but it's not the only one.

This triangle also contains the same measurements as the one in the sketch.

However, this triangle is not congruent to previous one.

Yes, these three particular measurements are the same, but not all measurements.

For example, this triangle contains an obtuse angle whereas the previous one didn't.

So let's check what we've learned.

You've got three tools on the left hand side.

Match the tools with what they're useful for when constructing a triangle.

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

Let's take a look.

When drawing the first side of a triangle, where the direction doesn't matter, we can do that with a ruler.

When we are considering all the possible positions for a side when drawing a second or third side at a given length, we can do that with a pair of compasses or compasses.

And when we are drawing at an angle to show what directions the sides on either side of an angle we take, we can do that with a protractor.

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

This task contains one question, and here it is.

Use pencil, paper and any other appropriate apparatus to make an accurate drawing of each triangle, based on the measurements given in each question.

They are parts a, b, c, and d.

In some questions, it is possible to construct two incongruent triangles.

And in these cases, do draw both triangles.

In some questions, you can only draw one triangle.

That's fine, just draw that triangle.

In some questions, it's impossible to construct any triangles.

And in these questions, draw the construction lines and explain why it's impossible.

And then once you've done that, if you've previously learned about the sine rule for non-right angle trigonometry, then you could use that to calculate the size of the missing angles in each of your triangles for parts a to d.

And then you can check your drawings by measuring those angles with a protractor.

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

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

Part a, it's only possible to construct one incongruent triangle and it looks something a bit like this.

Yours might be in a different orientation and that's fine.

If you worked out the size of the missing angles, you would've found out that one of them is 24.

6 degrees and other one is 125.

4 degrees when both rounded to one decimal place.

For part b, it's possible to construct two different incongruent triangles, and it would look something a bit like this or in different orientations.

If you wanted to work out the missing angles, you'd get these answers here.

For the one on the left, you'd get 48.

6 degrees and 101.

4 degrees.

For the one on the right, you'd get 131.

4 degrees and 18.

6 degrees.

Looking at the angles, we can definitely say now that these triangles are not congruent to each other because they do not have the same angles.

For part c, it's only possible to construct one incongruent triangle.

Yours might be in different orientations and that's absolutely fine, but it would still be congruent to this one.

You work out the missing angles, you'd get one angle as being 90 degrees, and other one as being 60 degrees.

And then in part d, it's impossible to construct the triangle given those measurements.

And by comparing to part c, you can see why.

In this case, one explanation could be that there is no point where the arc and ray line intersect.

It could also be explained using trigonometry by setting up an equation and showing that it has no solutions.

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 an angle that is not between them in the configuration side, side, angle.

The first side can be drawn with a ruler.

The angle should be drawn accurately with a protractor.

And the possible positions of the second side can be found with a pair of compasses.

Now, when you do this, sometimes you'll only be able to create one incongruent triangle, but sometimes you are able to construct two triangles with these same measurements.

However, all triangles with these measurements in the same place are only congruent when the unknown measurements are the same as well.

Well done today.

Have a great day.