Video: Pack 3 • Paper 1 • Question 20

Pack 3 • Paper 1 • Question 20

03:08

Video Transcript

The shape A is shown below. Enlarge shape A by a scale factor of negative two about the origin. Label the shape B.

We’re asked to perform an enlargement where the center of enlargement is the origin. The scale factor for this enlargement is negative two. This means that the distance from the center of enlargement, the origin, to each vertex on the new shape will be twice as much as the distance from the origin to the corresponding vertex on the original shape.

However, as the scale factor is negative, this means that the image will be on the opposite side of the origin to the original shape. We’ll take each vertex of shape A in turn.

To get from the origin to this vertex that I’ve marked in pink, we need to move one unit to the left and one unit up. To get from the origin to the corresponding vertex on the image, we need to double these distances but also reverse them. So instead of one unit left and one unit up, we now go two units right and two units down.

To get from the origin to the vertex that I’ve marked in green on the original shape, we go two units left and one unit up. To find where this vertex will be on the image, we therefore double the distances and reverse them. So we go four units right and two units down.

For the final vertex of the original triangle, we get there from the origin by going two units left and two units up. So the position of this vertex on the image is found by going four units right and four units down from the origin. We can then connect together the three vertices for the new triangle to give shape B.

Notice that the lengths in the triangle have indeed doubled. The horizontal side in shape A is one unit, and it’s two units in shape B. And the vertical side in shape A is also one unit, and it’s also two units in shape B.

Describe fully the single transformation that would map shape B onto shape A. The keyword here is “single.” We must write down just one transformation that would achieve this mapping. We can’t write down a sequence of transformations.

To get from shape B to shape A, we just need to reverse what we did to shape A in the first part of the question. This means that we want to perform an enlargement about the same point, the origin, but what’s the scale factor?

Well, it still needs to be negative, as shapes A and B are on opposite sides of the origin. But this time, we want to halve all of the lengths rather than double them. The scale factor that will achieve this is negative one-half. So the single transformation that will map shape B onto shape A is an enlargement about the origin, with a scale factor of negative one-half.