Lesson video

Hello everyone.

Thank you all so much for joining me, Mr. Gratton, in this lesson on constructions.

Grab a pair of compasses, a protractor, and a ruler for this lesson where we will look at the properties of rhombus, and use these properties to help us construct a bisector of an angle.

Pause here to have a quick look at some of the definitions that we will need today.

Let's start by constructing a rhombus when we're already given two of its sides.

Okay, here we have a rhombus.

Remember, a rhombus has four sides of equal length.

If a diagonal of a rhombus is drawn, then the rhombus is split into two isosceles triangles.

Notice that the two angles that were 114 degrees have been split and now halved to become 57 degrees each by that diagonal.

That diagonal on the rhombus becomes the base side that is shared by each of the two congruent isosceles triangles.

This base side is usually different in length to the other two sides of the isosceles triangle.

However, there are specific occasions when the diagonal is equal in length to the sides of the rhombus.

This means that we have two equilateral triangles instead, not two isosceles triangles.

Right, here's a quick check.

Here's a rhombus made from two isosceles triangles.

Pause here to find the sides of angles A, B, and C.

For A, we look at the isosceles triangle and create the equation 2a + 126 = 180 because those two base angles are equal.

Solving this equation gives A equals 27 degrees.

For B, this angle is the same on both copies of the rhombus at 126 degrees.

No calculations necessary there.

And C is twice the size of the base angle of the isosceles triangle, therefore, double 27 degrees is 54 degrees.

Right, here we have a diagram that shows two out of the four sides of a rhombus.

Clearly the other two sides are missing.

The angle between these two visible sides is 64 degrees.

We can call these line segments the legs of the angle, 64 degrees.

Any two lines or line segments that meet at a point can be called the legs of the angle that they create.

However, these two legs are sides of a rhombus, therefore, these two legs must also be equal in length.

Using a ruler, I measure each leg to be exactly seven centimetres long.

We can construct the other two sides of this rhombus by treating the rhombus as two congruent isosceles triangles with this shared base side.

Right, let's take a pair of compasses and set its width to seven centimetres by placing the needle on the open endpoint of a leg, point A in this diagram, and the pencil end on the vertex of the angle, point B in this diagram.

Then we begin to draw a full circle with a centre at point A.

Notice how every single point on the circumference of this circle is exactly seven centimetres away from point A.

This is absolutely important to remember.

Okay, let's construct another circle by placing the compass needle onto point C, which is the other open endpoint at the other leg.

Once again, we set our compasses to a seven centimetre width.

With the compass needle firmly placed on point C, let's draw a second circle.

Every point on the circumference of this second circle is also seven centimetres away from point C.

Therefore, both circles are congruent to each other.

However, Laura asks a very good question, how do these two circles help us complete the rhombus? Pause here to think about or discuss how these constructions can help us complete this rhombus.

The two circles intersect at two points, at point B, the vertex of the angle, and here at point D.

Drawing full circles is great because we can see the whole construction and understand all the points of intersection.

But full circles take up a lot of space and are unnecessary.

It is also possible to find point D by constructing only circular arcs with a radius of seven centimetres, rather than full circles.

However, you'll need to know roughly where the arcs intersect in order to find the exact point of intersection.

If you're in any doubt, make your arcs large to begin with to guarantee that point of intersection at point D.

No matter whether we use full circles or just arcs, these two sides are always going to be seven centimetres long, and these two distances are also seven centimetres.

Point D is seven centimetres away from both points A and C.

Therefore, the point of intersection at D is the final vertex of this rhombus, as it is seven centimetres away from the other two vertices of the rhombus, points A and C.

The opposite angle to 64 degrees is also 64 degrees.

For this check, we have a diagram that shows a partially completed rhombus.

Two sides of equal length are drawn.

Pause here to write down the distance between the tip of the pencil and the tip of the needle on this pair of compasses.

All sides of a rhombus are equal in length, so the compass width is 6.

4 centimetres as well.

Next up, pause here to write down the length of the radius of this circle.

The radius of this circle as well as a second circle with centre at the other open end point of the other leg will have a radius of exactly 4.

5 centimetres.

Next up, Sofia has attempted to construct a rhombus, but her construction has gone wrong somewhere.

Pause here to identify the correct reasons for what Sofia has done wrong when trying to construct this rhombus.

The compasses have not been set to the same width for each circle, and therefore the circles themselves are not the same size and they should be the same size.

Some good advice.

If you think your pair of compasses has slipped, always go back and check.

You can reset the compass size by placing the needle and pencil ends back onto a side of the rhombus before you begin drawing another circle.

Great stuff.

Thank you all for your attention so far.

Onto these two practise questions.

For question one, you are given two pairs of partially completed constructions.

For each construction, you are given two sides of a rhombus, complete both rhombi.

And for question two, you are given two sides of a rhombus.

Construct the rest of the rhombus and then draw on both diagonals.

Use a protractor to check that both diagonals meet at a right angle.

Pause now for these two questions.

Great effort on these first few constructions.

Pause here to compare your constructions to these on screen.

So far we have finished constructions of rhombi if given two equally long legs, which are also two sides of that rhombus.

However, can we still construct a rhombus if we're given two legs of an angle where each of the two legs are different in length? Well, let's have a look if we can.

Jacob here claims that it is impossible to construct a rhombus from these two legs, as both legs are different in length.

However, Lucas thinks it is possible.

All we need to do first is figure out a way to make the two legs of the angle the same length.

Jacob actually agrees with this and notices that we can use a ruler to extend the shorter line, make both legs the same length.

Whilst Jacob is technically correct, we can use a ruler to extend a leg to make it the same length as the other leg, sometimes lengths are incredibly tricky to measure precisely using a ruler.

For example, this leg is exactly how long? I'm not quite sure and neither is Jacob.

Jacob uses a ruler to extend this leg, like so, but Jacob does not think he drew it the same length, so this shape cannot be a rhombus.

Instead of using a ruler, what other maths equipment could we use to guarantee that the two legs become equal in length? Pause here to think about or discuss possible strategies that we could use.

Rather than using a ruler, maybe we can use a pair of compasses instead.

And Lucas agrees with this.

A circle has the same radius all around its centre, so drawing a circle with its centre at the vertex of the legs, the vertex of the angle will help us create two legs of the same length.

Let's have a look as to how.

And here's how.

Let's have a look at step one.

Let's place the compass needle on the vertex of the angle, and set its length to any length that is shorter than the shorter of the two legs of the angle.

And then we draw a full circle like so.

If we look at only the part of each leg that is inside of the circle, then they are both equal in length.

In this case, they're both 5.

5 centimetres long.

Lucas asks a good question, what if we made the length of the compasses larger? Would this still work in making the two legs equal in length? Well, let's have a look.

Let's make the compasses longer in width.

Ah, if we make the pair of compasses too wide, then one of the two legs may not be long enough to reach the circle.

We could extend the line segment, so the extended line segment intersects with the circle.

But why bother when shortening the compasses is easier and is one less step? So we can make the width of the compasses smaller than the legs of the angles.

Any of these widths will do.

However, this final compass width isn't great as we also have to extend the length of the leg, but this might be necessary if one of the two original leg lengths is really, really small.

The absolute best advice is to make the compasses as wide as the shorter of the two legs if it can reach that far, like so.

This circle is absolutely perfect in size.

This circle is big enough to use, but still small enough that it intersects with both legs.

But why do we do this though? Well, because the two parts of the legs that are inside the circle are two of the sides of the rhombus.

So we can just ignore the useless part of the leg that is outside of the circle.

We can represent the unnecessary part of the leg by either a dashed line segment or a faded out line segment.

However, do not rub it out completely.

And remember it can be very helpful to construct only two circular arcs, like this, rather than the full circle.

Now we have two legs of equal length, we can construct a rhombus just like before.

Place the compass needle on the open endpoint of one of the two legs and place the pencil on the vertex of the angle and then draw a circle.

We then repeat this by placing the compass needle on the other open endpoint of the other leg, and then we draw another circle like so.

We can also use two circular arcs rather than two full circles, like so.

That point of intersection is the fourth vertex of this rhombus.

Remember, this part is ignored.

It is not part of the final rhombus and that's okay.

Right, let's check your understanding.

Aisha begins to construct a rhombus from this angle like so.

Pause here to identify the point that is the final vertex of that rhombus.

Point B at this intersection is the final vertex of the rhombus.

For this check, we do not want to extend any leg when constructing this rhombus.

Pause here to consider the maximum radius of each circle in the construction of this rhombus.

12 centimetres is that maximum radius.

The maximum radius of each circle should be the length of that shorter leg of the original angle.

Any radius smaller than this is possible.

Whatever works best for the length of your pair of compasses.

Here is the completed rhombus.

Right, I will model good construction practise on the left.

We will construct a rhombus from a pair of line segments that looks like this.

Pause here to draw two line segments of your own of different length that meet at a common vertex.

You may have constructed something that looks like this.

Okay, firstly, place the compass needle on the vertex of the angle and set it to any sensible length shorter than the shorter leg, like so.

However, drawing a full circle may take up a lot of space, like this.

And sometimes it'll go off the page as well.

So keeping the compass needle in place, let's draw two small arcs instead, like so.

These two arcs go in the area where the pencil overlaps the two legs of the original angle.

Pause here to try this for yourself with two arcs.

Your two arcs may have looked like this.

So now we should have two intersections where each of the two intersections are two of the vertices of our rhombus.

Make sure to keep your compass width exactly the same or reset it to the same width as before.

And then place your compass needle on one of those two intersections and create an arc in this general area, like so.

Remember, if you are unsure of the location, make your arc longer.

This rough space is important as we expect the final arc that we are about to draw to intersect somewhere in this space.

Pause now to try constructing an arc on your construction like this.

Okay, yours could look like this.

Don't worry if the arc is a little longer than mine.

Next up, let's do the exact same again with the compass needle on the other intersection that you drew.

Notice how my needle is now on the horizontal leg.

With the compass needle in place, let's construct our fourth and final arc, one that intersects with the previous one that we just drew.

Again, we do this in that general area between our two line segments.

The end result are two intersecting arcs that look like this.

Pause now to try this for yourself.

Yours should look a little bit like this.

If not, remember, you can always go back and make your arcs a little bit longer until those two most recent arcs intersect each other.

The single point where the two arcs intersect is the final vertex of our rhombus.

Join this vertex to the two open endpoints to complete the rhombus.

Pause now to complete your own construction.

And a very well done if your rhombus looks anything like this.

Right, for this nearly completed construction, pause here to identify which of these arcs was first to be drawn.

A and C were the first to be drawn.

This is because they are the two arcs that cut off unnecessary parts of the angle's two legs.

This means that the two legs that remain are now equal in size.

Great stuff.

Onto the next set of practise questions.

For question one, pause here to complete the constructions for each of these four rhombi.

And for question two, here we have a pentagon.

We can construct a rhombus using two sides of the pentagon as the first two legs of our construction.

Pause here to construct two different rhombi, one with a side length of five units and an angle of 47 degrees, and the second rhombus must also have a side length of five units but not have an angle of 47 degrees.

Great effort so far.

Pause here for the solutions to question one.

And pause here for the correct construction for question 2A, as well as one possible construction for question 2B.

And pause here once more for some alternate solutions to 2B if you extend the length of some sides of that pentagon.

Right, last up, let's use a rhombus that we will create to bisect an angle.

Let's find out how.

Laura here is now happy that she can construct a rhombus, but why should she care? Why is it helpful? Well, let's have a look.

The diagonals of a rhombus intersect at right angles.

If a diagonal is drawn, then the interior angle of a rhombus is halved.

In other words, the angle is bisected.

This 70 degree angle will be bisected into two 35 degree angles.

Here we have a rhombus with its diagonals drawn.

Pause here to calculate the values of the angles A, B, C, and D.

A will be half of 124 at 62.

B will be supplementary to 124 at 56.

C will be half of 56 at 28.

And D will be a right angle at 90 degrees.

Right, let's bisect this angle.

This can be any angle, and the two legs can be of any size and different in length from each other.

A bisected angle will cut this angle exactly in half, so this 84 degree angle will become two 42 degree angles.

And in order to bisect any angle, we can just construct a rhombus, then draw on its diagonal, which we know bisects its interior angles.

So let's do exactly the same as before.

Set the pair of compasses to any width shorter than the two legs, and draw two arcs that intersect these two legs.

And then from these two arcs, construct two more in this general direction, making sure your compass width never changes.

Now we have the fourth vertex of our rhombus.

Draw on the final two sides of our rhombus, and draw on its diagonal to bisect the angle.

However, we can make our construction method a little bit more efficient.

It's not necessary to construct the full rhombus, even though we've got all of the construction lines to draw one.

All you need to do is draw a line from the point where the two arcs intersect each other to the vertex of your angle.

This will bisect your angle, like so.

Aisha has begun to construct a rhombus to bisect this 78 degree angle.

Pause here to identify which of these could be the next sensible step in Aisha's construction.

Aisha could very sensibly ensure that her compass width is still exactly the same as for her first two arcs, or she could jump straight into drawing two more arcs with the compass needle at the two current intersections if she's confident that her compass width has not changed.

However, if in doubt, it is always good to reset the length of your compasses so that the construction continues to make a rhombus and not a different quadrilateral.

Okay, last check.

Sam only wanted to bisect this 78 degree angle.

Pause here to identify which of these construction lines did not need to be drawn.

The last two sides of a rhombus are not necessary to be drawn, as the point where the two arcs intersect each other can be joined straight to the vertex of the angle in order to bisect it.

Amazing.

Let's put all of that construction into practise.

For question one, pause here to complete each construction of an angle bisector, and then use a protractor, but only at the very end, to measure the bisected angle.

You should have two equally sized angles.

Okay, for question two, here we have two reflex angles that need bisecting.

How does our method change, if at all, to adapt to these reflex angles? And for question three, construct this rhombus from only its description.

You can use a protractor but only for the initial construction of the 70 degree angle.

Then using trigonometry, calculate the precise lengths of the diagonals of your rhombus.

Pause here for these two questions.

And finally, question four, let's look at an incenter of a triangle.

This is the point where the three angle bisectors of a triangle intersect each other.

Find the incenter of each of these triangles.

And then for part B, follow these instructions to use your incenter to construct a circle.

What do you notice about that circle? Pause here for the final question, question four.

Amazing effort, everyone.

Pause here to compare your constructions for question one to these on screen.

For question two, your constructions will not have changed one bit.

All you need to do is bisect the conjugate angle to the reflex one, and then extend that line of bisection to visually bisect the reflex angle instead.

Pause here to compare your constructions to these on screen.

Onto question three, the two diagonals of the rhombus that you constructed should have been approximately 6.

9 centimetres and 9.

8 centimetres long.

Pause here to compare the precise answers found with trigonometry to your own.

And finally, question four, pause here to check the locations of the incenters for these triangles.

And as the name suggests, the incenter of a triangle is actually also the centre of a circle.

In fact, this circle is special because it is the largest possible circle that can fit fully inside of that triangle.

Great practical understanding, everyone, in a lesson where we have used the fact that a rhombus has four equal sides in order to finish constructing a rhombus from two of its sides, or from two legs of any length.

These constructions could use either full circles or just circular arcs.

Constructing a rhombus helps bisect an angle.

And lastly, for all of you amazing people out there, we looked at an incenter of a triangle as the single point where three angle bisectors intersect.

Well done everyone for your incredible effort today.

I have been Mr. Gratton, and you have all been superstars.

Until our next maths lesson together, take care everyone, and good bye.