Shape A is shown. Kerrie enlarges shape A by a scale
factor of negative two with a center of enlargement zero, zero and calls this shape
B. Darius rotates shape A by 180
degrees anticlockwise about negative 1.5, negative 1.5 and calls this shape C. Which coordinate is a vertex of
both shape B and shape C?
In order to be able to answer this
question, we first have to apply both of the transformations to our shape A. To complete an enlargement about
the center zero, zero, we first need to draw our rays. That is, we draw a straight line
connecting each vertex with the center of enlargement. Here, the center of enlargement is
negative, which means the shape is going to come out of the other side of the
center. The vertices of our enlarged shape
must lie somewhere along these lines.
Now, we’re given that the scale
factor is negative two. This means that the distance from
our center to our new shape must be twice the distance from each vertex on the old
shape to the center. Let’s start with this one.
To get back to the center of
enlargement, we can go three down and three units across. This means that we need to go six
units across and six units down to get to the relevant vertex on our
enlargement. That’s here.
Now, for this vertex, to get back
to the center, we go six down and three across. For our corresponding enlarged
vertex then, we need to go six across and 12 down.
Finally, this vertex, we need to go
three down and seven across to get back to our center. So to get to the enlarged vertex,
we’re going to go 14 across and six down. We now have the vertices of our
enlarged triangle B.
The next step is to rotate shape A
180 degrees anticlockwise about negative 1.5, negative 1.5. Tracing paper can be a really
useful tool to help us. By tracing shape A onto a piece of
tracing paper and putting our pen in the center of rotation, we can see that shape C
ends up as shown.
The coordinate that is a vertex of
both shape B and shape C is, therefore, negative six, negative six.