What is the image of the point
nine, eight under reflection in the straight line 𝑦 equals 𝑥?
These sorts of questions can be
quite difficult to visualize. So, let’s plot our point and the
line 𝑦 equals 𝑥 on a Cartesian plane. Here is our point nine, eight. Its 𝑥-coordinate is nine, and its
𝑦-coordinate is eight. The line 𝑦 equals 𝑥 is a diagonal
line. Every point on this line has equal
𝑥- and 𝑦-coordinates. For example, it will pass through
the point one, one; three, three; five, five; negative two, negative two; and so
on. In fact, it looks a little
something like this.
We’re going to reflect our point in
this line. So, we look at the perpendicular
distance from our point to the line. We can see that’s half of the
diagonal of one square. The image of our point will be the
exact same distance away from the line in the opposite direction. So, what are its coordinates? Well, they’re eight, nine. It now has an 𝑥-coordinate of
eight and a 𝑦-coordinates of nine. And so, we’ve worked out the image
of the point nine, eight under reflection in the line 𝑦 equals 𝑥. It’s eight, nine.
But we can, in fact, generalize
this. We take a point 𝑥, 𝑦. When we reflect it in the line 𝑦
equals 𝑥, it becomes 𝑦, 𝑥. In other words, the values of the
𝑥- and 𝑦-coordinates interchange. In general, we say that the
reflection of 𝑥, 𝑦 across the line 𝑦 equals 𝑥 is the point 𝑦, 𝑥. And the reflection of 𝑥, 𝑦 across
the line 𝑦 equals negative 𝑥 gives us the point negative 𝑥, negative 𝑦.