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
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
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
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
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.