Lesson video

Hello, there.

You made a great choice with today's lesson.

It's gonna be a good one.

My name is Dr.

Rowlandson and I'm gonna be supporting 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 right-angled triangle given the length of the hypotenuse and one other side length.

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 might want to pause the video if you want to remind yourself what they 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 use dynamic geometry software to draw triangles with perfect precision, and that way, we can use this to explore the theory behind the constructions and understand why they work.

Then in the second part of the lesson, we're going to replicate what we've just done, but using a pencil and some other appropriate 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 the information that is given to us, such as the length of the hypotenuse and one of a side length in a right-angled triangle.

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

The dynamic software that we'll be using in this lesson is GeoGebra.

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

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

That should open up a page that looks something a bit like this.

You've got a blank canvas on the right and you've got a toolbar on the left.

But there's only a small number of tools available so far, but you can click on More to access more tools and then you can scroll down to see the full range of tools available.

And if you still don't have the tool you need in there, you can click on More again to access even more tools, if necessary.

So here, we have a sketch of a right-angled triangle.

The hypotenuse has length 10 units and one of the other side length has length six units.

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

Let's start by drawing the shorter side that we know the length for.

To do that, find a section of tools under the subheading lines and 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'll type in six.

And that will draw a line segment that looks something a bit like this.

Now we want to draw a right-angle at one of those two points that we created.

Now to draw the right-angle, we should consider how to construct a perpendicular line at one of the two points.

Jacob says, "If we extend this line segment first, then we could use a similar technique to when we construct a perpendicular bisector." So let's do that then.

To extend the line, first, find the tools under the line subheading and click Line, and then click on each of the points from the previous line segments and that will create a line that goes in an infinite length in either direction.

Then Jacob says, "Now we want to mark two points that are equidistant from where the right-angle will be." So if we want the right-angle to be in the left of those two points, then we want to mark two points on either side of it that are equidistant from that point where the angle will go.

Like we can see on the screen here.

Jacob says, "That way, we can construct the perpendicular bisector between those points." And that perpendicular bisector will go through the point where we want it to go and create that right-angle.

So let's do that.

To mark the two points our equidistant away from where the angle would be, find the tools under circles and click Circle with Centre.

And then click on the point where you want the right-angle to go and somewhere else on the line.

And that'll create a circle that looks a bit like this.

We now want to mark two points on that circle that are also on the line.

We wanna mark where they intersect.

So under tools, find the points subheading and click on Intersect and then click on the circle and the line and it'll create two points where the circle intersects the line.

Those two points where a circle interceptor line are equidistant from the point where the right-angle would be 'cause the right-angle is at the centre of the circle.

Jacob says, "If we now construct a perpendicular bisector for the diameter, which is highlighted here on the screen, it will create a right-angle at the centre of the circle." So now we can think of this as a little smaller problem inside this bigger one.

And the smaller problem is to construct a perpendicular bisector for the line segment that's highlighted there on the screen.

Let's do that then.

So to construct a perpendicular bisector, we're first gonna click on Circles centre and radius.

We'll click on one of the two intersection points, such as the one the left, and choose any radius.

It doesn't matter what radius you use so long as it's long enough to go beyond the centre of the circle that is already drawn.

Like this.

And then repeat the same process on the other intersection point, choosing the same radius.

Like this.

Now we can see we have two circles that are of equal size and they intersect at two points.

One point is directly above where we want our right-angle to go, and the other point is directly below where we want our right-angle to go.

So we want to mark on those two intersection points.

Find the tools under points and click on Intersect.

Click on those last two circles that you made and that'll create those two intersection points.

And now we need to draw a line through those intersection points.

So in your tools, find lines and click the Line tool.

Click on those last two points that you made where the two circles intersect and it'll create a line that goes through the first line segment we drew, and it'll go through at that point on the far left of that line segment and it'll be at a right-angle.

Now there were quite a lot of steps to constructing that right-angle.

And if you were doing this on GeoGebra, there is a quicker way you could have done that.

An alternative method to draw on that right-angle on GeoGebra could be to go to the section of tools under the subheading of construct and click on the Perpendicular line tool.

Then click on the line segment and click on the point where you want the right-angle to go.

And that'll create a right-angle like this.

Now you might be thinking, why did we bother drawing all those circles previously to construct the perpendicular line? We could just have used the perpendicular line tool.

Well, the reason is because in the second part lesson, we're going to be looking to replicate some of this stuff with pencil and paper.

And unlike GeoGebra, we don't have a perpendicular line button, so we have to do things step by step.

So that's why we initially looked at how to draw it with circles because that's a process we'll be replicating later.

But going forward in this part lesson, we can use the perpendicular line tool.

So now we've drawn one side of our triangle and we have a right-angle faintly marked out.

Let's draw our hypotenuse.

We want it to be of length 10 units.

So find the tools under the circle subheading and click Circle: Centre & Radius.

Click on the point at the end of the short side, which does not have the right-angle because that's where we want the centre of the circle to be.

And then type in a length for the hypotenuse, 10 units in this case.

And that creates a circle that looks something bit like this.

We can't see the circle on our screen, but that's okay.

We just need the arc to intersect the line where the right-angle will be.

Every point along that arc is exactly 10 units away from the centre of that circle.

And two of those points meet the line we've drawn, which makes a right-angle with the shorter side, and they are at the points of intersection.

So let's mark those out.

So find the tools under points and click Intersect.

Click on the perpendicular line that you constructed and the last circle that you made, and that'll mark out the two intersection points.

And the distance between each of these interception points and the centre of the circle, which is at the point on the right of the first line segment here, those distances will be equal to the hypotenuse.

So to complete our triangle, find the tools under the polygon subheading and click Polygon.

Click on the points at either end of the first side and one of the points of intersection such as the one at the bottom.

And that creates a right-angled triangle that looks something a bit like this.

You could have used the other intersection and that would've created a congruent triangle.

Now Izzy says, "There are a lot of steps to construct that triangle, but the whole process is just a combination of lots of smaller constructions." For example, the right-angle was made based on how we would construct a perpendicular bisector.

Like this.

The hypotenuse is made based on the fact that every point on a circle is equidistant to its centre.

Like this.

So let's check what we've learned.

Here, we have Alex who is constructing a right-angled triangle and has drawn a line segment for one of its shorter sides.

Which tool can be used to construct the perpendicular line through one of its points using GeoGebra? Pause by choosing from A, B, C, or D and press play when you're ready for an answer.

The answer is C, perpendicular line.

Now you may have been tempted by B, the perpendicular bisector.

But if you use that on this line segment, the perpendicular line will go through the very centre of that line segment, the midpoint of it.

But we want the perpendicular line to go at the end of the line segment so it creates the right-angle at the end of where the side will be not in the middle of the side.

True or false.

Triangle, A, B, D and triangle, D, B, C in this diagram here are congruent.

Is that true or is it false? Pause while you choose and also choose a justification, then press play when you're ready for answers.

The answer is true.

And the reason why is that they are both right-angled triangles with a length of the short side and in the hypotenuse are the same in each.

They're congruent because you have your right-angle, hypotenuse and side length.

Now we've looked at one method for how to construct the right-angled triangle given the hypotenuse length and the length of the short side, but alternative methods are also available as well.

Izzy says, "This triangle could be constructed in a different way using a method based on a circle theorem." And that circle theorem is that the angle in a semicircle is always a right-angle.

Now if you've learned about circle theorems before, you should be able to make sense of the construction that we are about to do.

But if not, don't worry about it.

See if you can still replicate the process.

So let's start by drawing at the hypotenuse first this time.

To do that, we're going to find the tools under circles and click Circle: Centre & Radius.

Click on a point somewhere on the canvas for your first point and then type in half of the length of the hypotenuse and press Enter.

In this case, it'll be five.

And you'll draw a circle that looks something a bit like this where the diameter of this circle is the length of the hypotenuse.

Then find the tools under the line subheading and click Line and click in the Centre of the circle and somewhere on a circumference and it'll create something a bit like this and then find the intersect tool under points and click on the circle and the line to mark the points where the circle and line intersect.

Now the distance between the two points on a circumference is equal to the length of the hypotenuse, 10 units in this case.

So our diameter is acting as our hypotenuse.

So now we have the hypotenuse marked out.

Let's draw the shorter side, which we have a length for.

To do that, we want to find the Circle and radius tool.

Click on one of the two points where the diameter intersects the circle, and type in the length of the shorter side and press Enter.

And that creates a circle a bit like this.

Now the radius of the circle you just made is equal to the length of the shorter side, six units.

What we wanna see now is where these two circles intersect.

So under the point section of the tools, click Intersect, click on each circle and it'll create two intersection points.

Now the distance between the centre of the second circle we drew and each intersection point is equal to the length of the shorter side.

In this case, six units.

So to complete the triangle, we're going to use the polygon tool.

Click on the points at either end of the hypotenuse and on either of those intersection points that we made, and that'll create a triangle that looks a bit like this.

So how can we be sure that it's a right-angled triangle? Well, that's due to the circle theorem that angles in a semicircle are 90 degrees and the angle on the top of the triangle from where we are looking at it now that is 90 degrees because it is inside the semicircle on the left-hand side.

Therefore, it is 90 degrees and we have our triangle.

Now other triangles with the same measurements would be congruent.

So if we use the other intersection, it would produce a congruent triangle.

If we have plotted our second circle on the left intersection rather than the right intersection, that would create a triangle like this and that would also be congruent as well.

So let's check what we've learned.

What is the value of x in this diagram and justify your answer with reasoning? Pause while you do it and press play for an answer.

The answer is 90.

The angle in a semicircle is always a right-angle.

So now we know it's a right-angle.

If I tell you about the hypotenuse of the triangle is 12 centimetres, what is the length of the radius for this circle? Pause the video while you write it down and press play when you're ready for an answer.

The radius is six centimetres.

The hypotenuse is the diameter of the circle, so the radius is half the length of the hypotenuse.

And here we have a diagram that shows a right-angle of triangle constructed from two circles.

And what want you to tell me is what is the length of the hypotenuse? Pause video while you write it down and press play when you're ready for an answer.

The answer is 18 centimetres.

The hypotenuse is the diameter of this circle on the left there and that is double the length of the radius.

So 9 times 2 is 18.

In this same diagram, which angle is 90 degrees? You've got three to choose from.

Pause while you choose and press play for an answer.

The answer is angle CAB, and that angle is 90 degrees because angles in a semicircle are always 90 degrees.

Okay, it's all due now for task A.

This task contains one question and here it is.

Open up a web browser, go to geogebra.

org, and open up a blank geometry page.

And once you've done that, make an accurate drawing of the triangles that you can see represented as sketches.

You can use either method to construct them or you could always use one method for one and another method for another if you want to.

It's up to you.

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

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

For A, your triangle should look like this and you may have done it using either these two constructions.

Either way, these two triangles will be congruent.

For B, your answers should look like one of these.

For C, your answers should look like one of these.

All your answers could be in a different orientation and that's absolutely fine as well.

And for D, your answers should look like one of these or be in a different orientation.

You're doing great so far.

Now let's look to replicate what we've just done by drawing triangles using apparatus.

Here, we have a diagram that shows a right-angled triangle that was constructed using dynamic geometry software.

And we got Jun.

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

He says, "I have a pencil, ruler, and a pair of compasses." How could Jun draw the triangle accurately using this equipment? Pause the video while you think about this and press play when you're ready to continue.

Well, we can see all the construction lines in this diagram and there's quite a lot of them, so it does look quite complicated.

But Jun says, "We could break it down into a sequence of smaller constructions." He says, "I could draw the first side by measuring it with a ruler." And then he says, "I could use my ruler to faintly extend the line and this will give me room to construct a perpendicular line at one end of this side." He says, "This circle here, which is now highlighted, that creates points that are equidistant from where I want the right-angle to be.

I could draw this with my compasses." And then he says, "That if I construct a perpendicular bisector for this diameter, which is now highlighted on the screen, it would provide a right-angle for my triangle." The perpendicular bisector would create a right-angle in the middle of the circle, and that is where we want the right-angle to go.

He says, "These two circles, which are now highlighted, these are to construct the perpendicular bisector." And where those two circles intersect creates this perpendicular bisector here.

"And that is where the right-angle comes from.

This circle, which is now highlighted, that shows every point that is 10 centimetres from the other end of the side that I first drew, and this shows where the hypotenuse can go." So now we've looked at all the construction lines and understood how they've made the triangle we want to make.

Jun says, "I don't necessarily need to draw full circles when doing the constructions.

I could just use arcs that are long enough to show where the intersections will be." Like you can see on the screen here.

Right, now we've got to the bottom of that.

Let's try this with pencil, ruler, and a pair of compasses.

He says, "I could draw the first side by measuring it with a ruler." It doesn't necessarily matter which direction that first side goes in, but horizontal's always helpful.

And that's six centimetres.

Jun says, "I could use my ruler to faintly extend the line and this will give me room to construct a perpendicular line at one end of this first side." Like this.

He doesn't need to extend it in both directions, he could just extend it in one direction because he wants the perpendicular bisector to go through just one of those points on that first side.

He then says, "I'll use a pair of compasses to make two points on this line, which are equidistant to where I want the right-angle to go." So the needle for his compass is going to go at the point where he wants the right-angle to be and he draws a circle or two arcs that intersect that line.

And where those arcs intersect the line, those points are equidistant from the centre.

He says, "To create my right-angle, I'll construct a perpendicular bisector for the line segment that's between those two intersections." Like this.

And then we have our right-angle.

He then says, "To find where my hypotenuse could go, I'll open up my pair of compasses to 10 centimetres apart.

I'll put the needle at the end of the first side I drew and mark a point of the perpendicular line that is 10 centimetres away." Like this.

He says, "I'll use this intersection to draw the hypotenuse." So do it like this.

And now we've got pretty much everything we need to finish off our right-angled triangle.

"Finally, I'll join up the third side to complete the triangle." And there we go.

Well done, Jun.

So let's check what we've learned.

True or false.

When constructed a perpendicular bisector, the two circles that I can see here need to be the same size.

Is that true or is it false? And then choose a justification.

Pause while you do it and press play for an answer.

The answer is true.

And that's because using circles that are the same size cause the perpendicular line to intersect the midpoint of the over line segment.

Here, we have a diagram now that shows the same right-angled triangle that was constructed on dynamic geometry software, but by using an alternative method.

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

He says, "I have a pencil, ruler, and pair of compasses." How could Jun draw the triangle accurately using this equipment? Think about step by step what would Jun do.

Pause while you think about it and press play when you are ready to continue.

Well, first, Jun makes sense of the construction he can see.

He says, "The triangles constructed by using two overlapping circles." He says, "The diameter of this circle, which is highlighted, is the hypotenuse of the triangle.

That means its radius is half the length of the hypotenuse.

And for this circle, which is now highlighted, the radius is the shorter side of the triangle, the shorter side with the known length." Which, in this case, is six centimetres.

He then says, "I know that the triangle's right-angle because of the circle theory in which states that angles in a semicircle are always a right-angle.

So let's now try this with a pencil, ruler, and pair of compasses." He says, "I'll mark a point for the centre of my first circle and then I'll open up my compasses to half the length of the hypotenuse and draw a circle." So you want the hypotenuse to be 10 centimetres and that'll be the diameter of the circle.

So we open up our pair of compasses to five centimetres because that's the radius and draw a circle.

He says, "The diameter of this circle is the hypotenuse of my triangle, so it's 10 centimetres." He then says, "I'll open up my per compasses to the length of the shorter side." So at six centimetres.

"Then I'll put the needle at one of the points where a diameter intersect a circle and draw another circle." Like so.

So now we have a diameter, which is acting as the hypotenuse to that triangle.

We have a circle whose radius is the length of the shorter side, and we have two points where those two circles intersect.

So Jun says, "Finally, I'll complete my triangle by connecting one of the intersections with each end of the hypotenuse." Like this.

And now we can see we have our right-angled triangle, we have the correct length and we know its a right-angle because that angle is in a semicircle and these angles in a semicircle are always 90 degrees.

So let's check what we've learned.

Here, we have two diagrams. They show the constructions for two congruent triangles.

Now on the constructions on the left, you have some measurements given to you, but on the one right, you don't.

For the diagram on the right, what is the radius of each circle? You'll need two answers, one for each circle.

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

Well, here are your answers.

The circle on the left, the diameter of that circle is the hypotenuse, so it's radius will be half the hypotenuse, which is 7.

5 centimetres.

For the circle on the right, its radius is acting as a shorter side, that right-angled triangle, so its radius would be eight centimetres.

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

This task contains one question and here it is.

Take a sheet of plain paper, pencil, ruler and a pair of compasses, and then make an accurate drawing of each of triangles represented as sketches on the screen.

Now you can use either of the methods you've learned in this lesson to do that, or if you want to, you could use one method for one and another method for another and compare.

Then once you've drawn your four triangles in part E, you need to calculate the missing side length and the missing angles for each triangle rounding any non-integer answers you get to one decimal place.

Because these are right-angled triangles, you could use either Pythagoras' theorem or trigonometry or maybe a bit of both to do that.

Then once you have, you can check the accuracy of your drawings by measuring that side with a ruler and measuring the unknown angles with a protractor.

Pause while you do it and press play when you are ready for answers.

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

For part A, your triangles might look like one of these on the screen here or they may be in different orientations.

For B, they'll look like these on the screen or they might be in different orientations.

For C, do it look like these? And for D, do it look like these? Then for part E, need to go back to each triangle and work out the missing side length and missing angles.

Well, for the first triangle, here's your missing side length, it's 5.

3 centimetres and this is how you do it using Pythagoras' theorem.

You could then use trigonometry to work out one of the unknown angles and then subtract the 90 degree angle and the angle you've worked out from 180 degrees to work out the other unknown angle.

That would give you 41.

4 degrees for one angle and 48.

6 degrees for the other angle.

You may have worked these out in a different way, but hopefully you've got the same answers.

And then for second triangle, your missing side length is 12 centimetres, your missing angles are 67.

4 degrees and 22.

6 degrees.

And here's a suggestion how you could work it out.

Again, you may work it out in a different way.

For the third triangle, your missing the side length is 2.

9 centimetres.

One of your missing angles will be 15.

5 degrees, the other will be 74.

5 degrees.

And then for E, your missing the side length is four centimetres.

One of the missing angles is 22.

4 degrees and the other is 67.

6 degrees.

Fantastic work today.

Now let's summarise what we've learned.

An accurate drawing of a right-angled triangle can be made when the length of the hypotenuse and another side is known.

The shorter side can be drawn accurately with a ruler.

The perpendicular through the endpoint of the drawn side should be constructed.

You can do that using a pair of compasses.

The compasses should be used to indicate the potential positions of the hypotenuse as well.

And then you've got all the pieces you need to construct your right-angled triangle.

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

Well done today.

Have a great day.