### Video Transcript

A rectangle is drawn on the grid.
a) The rectangle is reflected in the line 𝑦 equals three. Circle the number of invariant
vertices of the rectangle. b) The rectangle is enlarged by a scale factor of two
with a centre of two, two. Circle the number of invariant
vertices of the rectangle. c) The rectangle is translated by the vector one,
two. Circle the number of invariant
vertices of the rectangle.

Let’s begin by recalling what we
mean by the word invariant. Invariant means unchanged. For each question, we’ll need to
consider the number of vertices — remember those are just corners of the shape —
that remain unchanged after the transformation. For part a, we’re going to reflect
the triangle in the line 𝑦 equals three. This is the line for which all of
its coordinates have 𝑦-values of three. It’s a horizontal line that passes
through the 𝑦-axis at three.

To reflect the rectangle in this
line, we’ll reflect each individual vertex. Notice that two vertices — that’s
the one at two, three and six, three — lie on this line. They’ll actually remain where they
are. The vertex at two, two though is
one square away from the mirror line. Its image will also be one square
away on the other side of the line; it’s here. Similarly, the vertex at six, two
is one square away from the mirror line. Its image will end up one square
away from the mirror line on the other side; it’s here. The image of the reflected
rectangle is as shown. We can see that there are exactly
two invariant vertices of this rectangle. They lie at two, three and at six,
three.

Now let’s consider part b. To enlarge this shape by scale
factor of two about the centre two, two, we’ll work out how far away each vertex is
from the centre and then multiply that by two. Notice that one of the vertices
lies on the centre itself. Since the distance from the centre
here is zero, when we double this distance it will still be zero. So this point remains
unchanged. This vertex is invariant. We should be able to spot that only
one of the vertices will be invariant. But let’s see where the rest end
up.

The point two, three is one square
away from the centre. When we enlarge this shape, we need
to double it. And in the same direction, we need
to be two squares away from the centre; that’s here. The point six, two is four squares
away from the centre. When we double this, we need to be
eight squares away from the centre in the same direction. That’s here: 10, two. We can now spot that the enlarged
shape is as shown. And again, we can see that the
number of invariant vertices is one.

For part c, we’re going to
translate the rectangle by the vector one, two. We need to move it one space right
and two spaces up. The entire shape slides in that
direction. This means that none of its
vertices can be invariant. It’s zero.