Question Video: Determining the Position of a Point after Reflecting in a Given Straight Line given the Point’s Coordinates Mathematics • 8th Grade

Video Transcript

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.