Lesson video

Hi, I'm Mr. Bond.

And in this lesson, we're going to be drawing loci from shapes.

Before we start thinking about loci from a shape, let's remind ourselves what the word locus means.

A locus is a set of points that follow a particular rule or pattern and loci is the plural of locus.

So for example, if I wanted to find the locus of the points, two centimetres from the rectangle ABCD, I could do it like this.

We'll start at point A and we'll measure perpendicularly two centimetres from the line segment AB, like this.

And we can mark a point.

If we did this all the way around the outside of the shape, we'd end up with points, looking like this.

Now if we connect to those points, let's have a look at the properties of the locus.

If we connect these points at the top, we have a line segment, the same length as AB and parallel to AB that's two centimetres away.

And this would happen for all of the line segments, BC, CD and AD Now let's think about the curved parts.

These would be quarter circles.

So this is the locus of the points, two centimetres from the rectangle ABCD.

Now let's think about how we'd actually draw this.

It's quite similar to drawing the locus of points from a line segment.

So in this example, we want to draw the locus of points that are four centimetres away from the triangle XYZ.

So we'll start by focusing on each of the points X Y and Z, starting with point X.

We know that the locus from a point always forms a circle, but this isn't just a point, it's a shape.

But still the arc around the point will follow a circular path.

So we'll start by drawing a circle that's four centimetres away from point X, like this.

We need to do the same thing at the other two verses, like this.

This will really help us to draw each line segment of our locus on.

We need to draw a line segments that are the same length is each length of the triangle and it's a tangent to each of the circles.

So that's like this, this, and this.

But of course, just like when we drew the locus of points away from a line segment, we need to rip out some parts of the circles.

These are the parts of the circles that are inside our locus of points, like this.

Here's some questions for you to try.

Pause the video to complete the task and resume the video when you're finished.

Here are the answers.

For the square and the triangle, you could have found the loci exactly the same way as the examples that I've shown you already.

It'd be slightly different for the circle though.

The most accurate way to do this would have been to place a compass point in the very centre of the circle and then open the compass two centimetres wider than the radius of the original circle.

And then draw on a new circle around the outside of the original circle.

Of course, making sure that you use a ruler to be as accurate as possible.

Here's another question for you to try.

In this question, remember to think about the scale.

Pause the video to complete the task and resume the video when you're finished.

Here's the answer.

So for this question, I asked you to take account of the scale, but it was quite a simple scale, really? One centimetre is the same as one metre in real life.

So we just needed to draw the locus of points, two centimetres from the edge of the rectangle in our diagram.

Here's another question for you to try.

Again pause the video to complete your task and resume the video when you're finished.

Here are the answers.

But this question, again, it being a circle, it would have been most accurate to place the point of a compass precisely in the centre of the circle and then open the compass so that its radius is one centimetre greater than the radius of the original circle.

Then draw the locus of points, then open it up so that it's two centimetres greater than the radius of the original circle.

Draw a second locus of points and then shade between the two.

Here's another slightly more complex example.

We want to draw the locus of points that one centimetre away from the hexagon.

So to start with, three each of these five vertices, we draw a circle with radius one centimetre, just like we did for the triangle, really.

And then between each of the circles, we draw a line segments that are parallel to the corresponding line segments of the hexagon and a tangent to the circles, like this.

Now, we'll need to rip out parts of the circle for each circle.

Those parts that don't form the locus, and that would look like this.

But as you can see, I've not drawn the whole locus of points.

What would happen at this part of the shape? I'm not going to give it away for now because you're about to try a question that's very similar.

Let's see if you can figure it out.

Here's that question for you to try.

Pause the video to complete your task and resume the video when you're finished.

Here's the answer.

Hopefully you followed exactly the same process as we did in the example that I've just shown you.

And then for the final part, hopefully you didn't fall into the misconception that there's some sort of arc in this final part.

The final part, is simply made up of two straight line segments.

That's all for this lesson.

Thanks for watching.