Video: GCSE Mathematics Foundation Tier Pack 3 • Paper 1 • Question 26

GCSE Mathematics Foundation Tier Pack 3 • Paper 1 • Question 26


Video Transcript

The shapes 𝐴 and 𝐵 are shown below. Fully describe the single transformation that maps shape 𝐵 onto shape 𝐴.

First, let’s just remind ourselves of the possible types of transformation that exist. This transformation could be a translation, a reflection, a rotation, or an enlargement. Notice that the question specifies that it must be a single transformation. So we must be able to map shape 𝐵 onto shape 𝐴 in a single step, not with a combination of transformations.

Now a translation is just a move or a shift of the original shape. So it remains the same size and in the same position or orientation. So this rules out a translation because shape 𝐵 and shape 𝐴 are in different orientations.

We can also rule out an enlargement as corresponding sides on shapes 𝐴 and 𝐵 are still the same lengths. Now it would be possible to have an enlargement with a scale factor of negative one. And this would keep the length the same. However, the two shapes would be upside down relative to each other. They aren’t upside down. Instead, it just looks like shape 𝐵 is on its side relative to shape 𝐴. So the transformation we’re looking for isn’t an enlargement.

This leaves reflection and rotation. Now reflection and rotation do both cause the shape to be in a different orientation, so they could still both be possible. However, if we were to reflect in a diagonal line somewhere between the two shapes, then the longest side of 𝐴 would be closest to the mirror. But on the shape 𝐵, the longest side would be furthest from the mirror. This means that the transformation we’re looking for is not a reflection either, as in the mirror image the same side would be closest to the mirror for both.

So the transformation that we’re looking for is definitely rotation. To fully describe a rotation, we need to give three further pieces of information: the angle of rotation; the direction of rotation, so that’s clockwise or anticlockwise; and the centre of rotation, which is the point that remains fixed.

If you were using tracing paper to perform a rotation, this is the point where you put the point of your pencil. Now the direction is probably the easiest of these three pieces of information to work out. If we’re going from shape 𝐵 to shape 𝐴, then this means we’re rotating in an anticlockwise direction. We can also see that shape 𝐵 has gone from being on its side to being stood upright, so it’s turned through one right angle, or 90 degrees.

Now we need to work out the centre of rotation. Now this is a little bit trickier. If you have access to tracing paper, you could try placing your pencil point in different places and then rotating the paper until you find a point which maps shape 𝐵 exactly onto shape 𝐴. Without tracing paper, we just have to draw each of our trials.

Now a sensible point to start with is perhaps the origin, the point zero, zero. So I’ve marked this with a cross, and I’ve drawn on this orange arm that connects the origin to one corner of shape 𝐵. Now what I can do is rotate this arm through 90 degrees anticlockwise. And what you’ll see is that it actually connects to the corresponding corner on shape 𝐴.

We could do the same thing with any of the other corners of shape 𝐵. So draw on the arm connecting the origin to each corner, then rotate that arm through 90 degrees. And what you see is that it does actually connect to the corresponding corner on shape 𝐴 every time. So this means that the point that we try out as the centre of rotation is correct. And we can say that the single transformation that maps shape 𝐵 onto shape 𝐴 is a rotation, 90 degrees anticlockwise about the origin.

You may also see the phrase “about the origin” written as “centre of rotation the origin” or perhaps instead of the word “origin” it could be given as coordinates of zero, zero. And either of these would be perfectly fine.

Finally, it’s just worth pointing out that there are actually two ways of describing a rotation, because we can rotate in different directions. So rather than anticlockwise, we could rotate in a clockwise direction. As there are 360 degrees in a full turn, we can find the angle of rotation in a clockwise direction by subtracting our angle of rotation in an anticlockwise direction from 360 degrees. So a rotation of 90 degrees anticlockwise is equivalent to a rotation of 270 degrees clockwise. And this would also be an acceptable answer. The centre of rotation, the origin, remains the same.

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