Lesson video

Hello everyone and welcome to another math lesson with me, Mr. Gratton.

In today's lesson, let's use the properties of a rhombus to construct an angle bisector.

Pause here to take a look at the definitions of bisect, rhombus and congruent.

First up, let's construct a rhombus using two of its adjacent sides and the angle between them.

A rhombus is so helpful in creating many different constructions, but Laura asks, what makes a rhombus so special? Well, here are some properties.

Property one opposite angles in a rhombus are always equal as shown by these angle markers.

Property number two, all four sides in a rhombus are the same length as shown by these hash marks and property number three, there are two pairs of parallel sides as shown by these parallel feathers.

Here's another property about a rhombus.

If you draw on the diagonal of a rhombus, the rhombus will be split into two congruent isosceles triangles.

The angle that the diagonal cuts through will have halved from 114 degrees to 57 degrees.

The diagonal that you draw is a base side that is shared between each of the two isosceles triangles.

Okay, two quick check questions.

Question number one, this quadrilateral is a rhombus, find the size of angle A.

Pause now to think about the properties of a rhombus and give this question a go.

The answer is 127 degrees because opposite angles in a rhombus are equal.

Question number two, this rhombus has a side length of 13 centimetres, find its perimeter.

Pause now to give this question a go and the answer is 52 centimetres because all of the sides in a rhombus are of equal length.

This isosceles triangle has been duplicated with the base sides of each isosceles triangle put together to create the rhombus that you see on screen.

What are the sizes of the angles A and B for this rhombus that has been created? Pause now to give this question a go.

Angle A remains in variant so it stays at 132 degrees.

However, angle B has been doubled from the 24 degrees in the isosceles triangle to 48 degrees as the angle in the rhombus.

This diagram shows two adjacent sides of a rhombus.

The angle between those two sides is 64 degrees.

A line segment can be drawn between its two open endpoint to create the base side of an isosceles triangle.

Constructing a second identical triangle from that shared base side will result in a rhombus.

These two angle legs are sides of a rhombus and because I know it's a rhombus, I know for certain that they are of equal length at seven centimetres each.

By setting these pair of compasses to the length of one of the sides of this soon to be rhombus, the compass width is set to seven centimetres.

By placing the needle end of the pair of compasses at point A, the open endpoint of one of its sides, we can then draw a circle that has a radius of seven centimetres.

Every point on the circumference of this circle is exactly seven centimetres away from point A, another circle with exactly the same compass width can be drawn by placing the compass needle onto point C.

The other open endpoint of the angle given, and again, every point on the circumference of this circle is exactly seven centimetres away, but this time from point C.

To answer Laura's question, I focus on the second intersection of these two circles at point D by connecting a line segment from each open endpoint of the angle leg to point D, a rhombus is constructed where point D is the final vertex of the rhombus seven centimetres away from both points A and points C.

As you can see here, because we have constructed a rhombus, I know that the opposite angle to 64 degrees is also 64 degrees.

Okay, here's a quick check of the method to construct a rhombus.

In this diagram, we have a partially completed rhombus with two sides of equal length drawn.

What is the compass width? The distance between the tip of the pencil and the tip of the needle on this pair of compasses? Pause now to give this question a go.

The answer is 6.

4 centimetres because these are both side lengths of rhombus.

They are both of equal length.

And here's another diagram of a partially completed rhombus.

What is the length of the radius of this circle? Pause now to give this question a go.

And the radius is 4.

5 centimetres.

Again, the radius of the circle needs to be the same length as the length of one is the sides of that rhombus and Sophia is sad, she's disappointed that she has not effectively constructed a rhombus.

Which of these statements correctly describes why Sophia's attempt at constructing a rhombus has gone wrong? Pause now to look through each of these sentences and choose the correct ones.

The correct answers are B and C.

Actually, they're the same answer because Sophia hasn't kept the compass width the same between both circles, both circles ended up being a different size.

If you think your pair of compasses has slipped, so the width has changed, always go back and check and resize them.

So both circles are the same length.

Okay, I will do a demonstration to show you how to construct a rhombus with the information on the left.

After each set of steps, perform those same steps for the rhombus that I've described for you on the right hand side.

Step one, use a ruler to draw a line segment of 4.

5 centimetres and then use a protractor to measure an angle of 70 degrees from one endpoint of the line segment that you just drew.

Try this for yourself.

Pause now to give this a go.

And here's what yours may have looked like for a side length of four centimetres and an angle of 50 degrees.

And use your ruler to draw another line segment of 4.

5 centimetres this time at an angle of 70 degrees from the endpoint of the line segment that you drew previously.

Pause now to try this for yourself.

Here's what your four centimetre line would've looked like.

What we've constructed so far are two sides of a rhombus with an angle of 70 degrees between those two sides, both legs of the angle of equal length.

Here's the next set of steps.

Place the compass needle onto the open endpoint of one side of your rhombus and place the pencil on the vertex of the angle and construct a full circle.

Try this for yourself and pause here to do so.

And this is the rough location of where your first circle should have gone.

And your next step is exactly the same as the previous one.

Draw another circle by placing the compass needle onto the other open endpoint of your angle and the pencil end at the vertex of your angle.

Here is what your second circle should have looked like.

Try this for yourself by placing your compass needle onto the other open endpoint of your angle.

Pause to give that a go.

Here's your second circle and identify the intersection point of the two circles.

Because this is the final vertex of the rhombus we are going to construct, join up each open endpoint to the intersection to create your final rhombus like so.

And give this a go for yourself.

Pause now to do so.

And here's what your final rhombus should look like.

To check how close your construction is to a fully correct rhombus, each of the four lengths should be four centimetres, and the opposite angle should also be 50 degrees.

Okay, onto some independent practise.

Complete each of these constructions to create a rhombus.

After you have finished each construction, measure the size of the marked angles and check that the opposite angles are equal.

Pause now to do this for all three rhombi and for question number two, complete each construction to create these rhombi, pause now to do this and onto the answers for question one, pause now to have a look at these three rhombi.

And pause again to have a look at what these three rhombi should have looked like.

So far we've been able to construct a rhombus from an angle where both angle legs are the same length.

What if we had to construct a rhombus from an angle where both legs are different in length? Let's have a look.

Jacob says it's impossible to construct a rhombus from that diagram because the legs of the angle are different lengths, whereas Lucas says it's possible as long as you can make both the legs the same length.

Is Jacob's idea to extend the shorter line sensible? Well, maybe not.

Let's have a look why.

Jacob is unsure whether both of the line segments are of equal length, so he cannot guarantee that his construction will make a rhombus.

Jacob's second idea seems more effective using a pair of compasses to make both legs the same length.

And as Lucas says, this will work because a circle has a constant radius to construct a rhombus with an angle of 56 degrees.

Place the compass needle on the vertex of that angle and set its length to shorter than the shortest length of that angle.

Make sure that that the compass width is still a sensible size to reduce any inaccuracies later on.

From there, draw a full circle and Jacob is satisfied that the part of each leg that is inside the circle is of equal length to each other.

What if we had a compass width larger than one of the two legs? Well, we would draw a circle where one of the legs doesn't reach the circumference of that circle, and this would be a problem as the two legs of that angle would still be of different length.

And so it is possible to make the compass width smaller than both of the legs of the angle.

However, you should never make them larger than either of the two legs.

And so Lucas's final assessment is pretty accurate because it is always possible to set the compass width to exactly the length of the shorter leg because we only care about the parts of the legs inside of the circle because we've made them the same length, we can just ignore the useless part of the legs that are outside of the circle.

Now that we've got two legs that are equal length, our method is exactly the same as before.

Placing the compass needle at one of the open endpoints and drawing a circle and then doing the same again at the other open endpoint to create two intersecting circles that make a rhombus for each of the three circles that you draw in this construction make sure that the compass width is exactly the same for all three.

Do note that the part of the original leg that is outside of the circle isn't included in the final rhombus, and that is okay.

And for the second demonstration, pause here to draw any two line segments of different length.

And here's one example of what you could have done, but literally it could have been any two connecting line segments to construct a rhombus, place the compass needle on the vertex of the angle and place the pencil on the open endpoint of the shorter of the two legs.

And from there, draw a full circle.

Pause now to try this yourself for your construction.

And here's an example, as long as your compass needle was on the vertex of the angle and the pencil end on the endpoint of the shorter of the two legs, this first step of the construction will have been done correctly.

This part of the leg that is outside of the circle will not be used anymore through any of the construction or in the final rhombus, but do not rub any of your construction lines out.

And now the rest of this construction will be exactly the same as the construction that we saw in cycle one.

Both circles should have a centre at the point where each line segment intersects with the first circle that we drew and the pencil end at the vertex of the angle.

So the first circle should look like this.

The compass needle should then go here, and the second circle should look like this.

Pause now to try this yourself for your construction, making sure that the compass needle goes at the open endpoints at the places where the circle intersects the two legs of the angle that you drew previously.

And here's what yours may have looked like.

And the final step will be to identify the location where the two circles intersect, and to draw on the final two sides of your rhombus.

And pause now to try this yourself for your final rhombus.

And here's what your final rhombus could have looked like.

Here's a check from Aisha who is constructing a rhombus from this angle as we can see, from this construction, which point is the final vertex of her rhombus? Pause now to have a think, a final rhombus would look like this.

And so B is the correct answer.

In the construction of a rhombus from this angle, what is the maximum radius of each circle in your construction? Pause now to have a think.

And here are what the three circles should look like and therefore the maximum radius of each circle is 12 centimetres.

Remember, the maximum radius of each circle should be the length of the shorter leg in the angle.

Any radius smaller than the shorter two legs is possible.

Whatever works best for your pair of compasses.

Okay, onto the practise pause here to complete each of these partially completed constructions to draw a rhombus.

Pause here for question number two.

Again, to complete the construction, to draw a rhombus and onto the answers, your completed construction should have looked like this.

And for question number two, your constructions should have looked like this, to check use a protractor and measure the opposite angles of each rhombus.

If each pair of opposite angles is the same size, your construction is pretty accurate and a very well done in all of the constructions that you've done so far.

But can we make our constructions a little bit more efficient? Let's have a look.

Izzy is correct.

These constructions can get very messy very quickly, but Sam's observation is also correct.

The construction of a full circle is not necessary.

The only points on a circle that matter are the points where a circle intersects with another circle or a line segment like so.

If your construction is at its most efficient, only four circular arcs would need to be drawn for any rhombus.

Let's do one final demonstration.

Follow the steps that I show you when prompted.

Pause here like before, to draw any two line segments that intersect at their endpoints.

Here's an example of what you could have done.

Unlike with the previous demonstration, place the compass needle on the vertex of the angle and set it to any sensible length, shorter than the shorter leg.

By keeping the compass needle in place, use the pencil to draw two small arcs in the areas where the pencil overlaps the two legs of the angle like so.

Pause now to try this for yourself.

And remember, the compass needle stays firmly fixed on the vertex of that angle.

And here's what yours may have looked like.

Each intersection is a vertex of the rhombus, and therefore the next set of constructions that we are going to make will have to have circular arcs with the same radius as the ones you did previously.

To ensure this place your compass needle on one of the intersections and your pencil on the vertex of the angle to set the compass width to the same as the one that you had before, with your compass needle on one of those two intersections, create an arc in this general area.

We expect the two final arcs to intersect in this area.

And if you are unsure, make your arcs longer to begin with so you get a feel for the rough areas where the arcs will intersect.

And so because I expect my intersection to be in this general area, I draw an arc here, pause here to try this yourself.

And remember, if you are uncertain, make the arcs longer to begin with.

Here's what your arc could have looked like.

Now we do exactly the same thing again for the other place where the arc intersects with the length of your angle.

And so we draw another arc in the general location where it will intersect with the previous one that we drew.

And now try this for yourself.

Pause here to do so.

Your final arc should have been approximately here, but if your two arcs did not intersect, then go back and make your two arcs a little bit longer until they do.

Just like before, the location where the two arcs intersect is the final vertex of the rhombus.

Join this vertex with the two intersections between the legs of the angle and the two first arcs that you drew to create your rhombus.

Pause here to try this for yourself.

And here's what your final rhombus could have looked like.

And one quick check in this nearly completed construction, which two arcs were the first to be drawn? Pause now to think of your answer.

And the two first arcs to be drawn are A and C.

The first two arcs should have had the compass needle at the vertex of the angle, and the first two arcs should have intersected with both legs of that angle and for the practise of this cycle by drawing only arcs and straight line segments, complete the constructions of each of these rhombi.

Pause now to do so.

And here's what each completed construction would have looked like.

We've constructed a lot of different rhombi, but I wonder why, and Laura wonders exactly the same thing.

What is the purpose of constructing a rhombus? This is because the diagonals of a rhombus intersect at right angles.

If a diagonal is drawn, the interior angle that the diagonal passes through is halved.

The angle is bisected to be divided into two equal parts.

In this rhombus where the diagonals have been drawn, calculate the values of A, B and C.

Pause now to think about bisection and answer this question, angle A is the bisection of 124 degrees.

And so A equals 62 degrees.

Angle B is the bisection of 56 degrees, and so B equals 28 degrees.

And because angle C lies on the intersection between those two diagonals, the intersection is always at an angle of 90 degrees because they meet perpendicularly, the most efficient steps to bisecting any angle is exactly the same as the method to construct a rhombus.

And so to construct this line segment here, to bisect the 84 degrees, we need to construct a rhombus and then draw on its diagonal.

We follow exactly the same method as before.

Our first step is to construct two arcs that intersect with the two legs of this angle.

And from these two intersections, we construct two more arcs that intersect allowing us to construct a rhombus.

After constructing the rhombus, we connect the vertex with the angle 84 degrees on it to the location where the two arcs intersected to bisect our angle.

Alternatively, we don't even need to draw on the final two sides of our rhombus.

It is possible to draw a straight line from the point where the two arcs intersect through the vertex of your angle, and that line will have bisected the angle.

Okay, onto the final few checks, Aisha begins construction of a rhombus to bisect this angle of 78 degrees.

Which of these next steps could be a sensible one for Aisha to follow in her construction? Pause now to look through all of these options and choose the right ones.

The answers are A and C.

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

For Sam's construction, which of these construction lines did not need to be drawn? Pause now to have a think.

And the answer is C.

The last two sides of the rhombus do not need to be drawn.

Okay, onto the final set of practise Questions for question number one, use a protractor to measure the acute or the obtuse angle that is labelled and then complete the construction and measure the angle of your bisection to see whether it is exactly half the original angle given.

Pause now to do so.

And for question number two, bisect each of these given angles and measure the size of the bisected angle.

Pause now to do this final question and your answers for question one are as follows.

Pause now to see whether your measured angles and bisected angles match these.

And for question two, angle A has been bisected to get a value of 36.

The two bisected angles for B are 54 degrees and 37 degrees.

And for C, the two reflex angles have been bisected to get 153 degrees and 135 degrees.

Very well done if you've got those two answers.

Thank you so much for all of your hard work and lesson where we have constructed rhombi from angles with both equal and different sized legs and constructed rhombi more efficiently by using arcs rather than full circles.

We've also understood the purpose of constructing rhombi to bisect an angle.

That is amazing work.

Thank you so much for joining me, Mr. Gratin, in another math lesson, and I hope to see you soon for some more maths.

Have a good day.