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.

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 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 of 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 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 throughout 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.

Her 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 of 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 very well done on 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 gonna 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 leg 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 you 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 in the 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. Gratton, in another maths lesson.

And I hope to see you soon for some more maths.

Have a good day.