Reflect triangle 𝐴 in the 𝑦-axis
and then in the 𝑥-axis. Which triangle is its image?
Let’s begin by reflecting triangle
𝐴, that’s this one up here, in the 𝑦-axis, which is this vertical line. We know that when we reflect a
shape in a mirror line, the shape will end up exactly the same distance from the
mirror line, but on the other side. And it will be sort of flipped. And one way we can find the image
after reflection is to do this vertex by vertex.
Let’s begin with this vertex on
triangle 𝐴. We measure the perpendicular
distance of this vertex from the mirror line, and that’s two units. We now come the exact same distance
out of the mirror line on the other side. And when we do, we see that the
vertex is on triangle 𝐶. And so, we can probably deduce that
the image of triangle 𝐴 after its first reflection is triangle 𝐶. But let’s double-check this with
Let’s take this vertex at the top
of our triangle. Once again, we measure the
perpendicular distance of this vertex from the mirror line, and that’s three
units. We come out the other side of the
mirror line the exact same distance, and we end up with the second vertex at 𝐶. And in fact, if we did that with
the third vertex, we’d end up with the third vertex at 𝐶. And so, the image of triangle 𝐴
after the first reflection is triangle 𝐶.
But then we’re told this triangle
is reflected in the 𝑥-axis, and this is this horizontal line. And so once again, we know that the
image of 𝐶 after the reflection will be the exact same distance away from the line
directly at the other side. And so we measure the perpendicular
distance of our first vertex from the mirror line, and it’s two units once
again. We carry on out the other side of
the mirror line at the same distance, and we end up at this vertex at point two,
negative two. We can do the same with our other
vertices, and we end up with a vertex at point five, negative three and one at
three, negative five. And so, the image of 𝐶 after its
reflection is triangle 𝐷.
And so, we could therefore say that
when we reflect triangle 𝐴 in the 𝑦-axis and then in the 𝑥-axis, the triangle
that is its image is triangle 𝐷. Now, in fact, this wasn’t
necessarily the quickest way to achieve this. And this is because if we reflect a
shape in one axis and then reflect it in the other, that’s the same as rotating that
shape 180 degrees about the origin or the point zero, zero. If we look at 𝐴, if we rotate that
180 degrees about this point, we do indeed get to triangle 𝐷. Either method is perfectly valid as
long as we get the answer 𝐷.