Learn About Reflection Over the Line Y=X

In this video, you will learn how to do a reflection over the line y=x.

The line y=x, when graphed on a graphing calculator, would appear as a straight line cutting through the origin with a slope of 1.

When reflecting coordinate points of the pre-image over the line, the following notation can be used to determine the coordinate points of the image:
r_y=x=(y,x)

For example:

For triangle ABC with coordinate points A(3,3), B(2,1), and C(6,2), apply a reflection over the line y=x.

By following the notation, we would swap the x-value and the y-value.
A(3,3), B(2,1), and C(6,2) would turn into
A'(3,3), B'(1,2), and C'(2,6)

Video-Lesson Transcript

In this lesson, we’ll take a look at reflecting over a line $y = x$ .

$r_{y = x}$

First of all, what is a line $y = x$ ?

This is a line that for every $x$ value, we get the same $y$ value.

$y = x$ will look something like this. A diagonal straight line.

When $x = 1$ , $y = 1$ . When $x = 2$ , $y = 2$ . And when $x = 3$ , $y = 3$ . And so on.

If we have a point, for example, $(3, 1)$ , we’re going to reflect it over. We need to move perpendicular to it.

One side should be perpendicular to the other side.

Our image now is $(1, 3)$ .

All we did was switch the $x$ and the $y$ values.

For example, if the initial image is $(1, -4)$ .

We have to make a line perpendicular to measure the distance. Then make the same line distance on to the other side.

The reflected point is $(-4, 1)$ .

When we reflect over the line $y = x$ , we just switch the values of $x$ and $y$ .

$r_{y = x} (x, y) \rightarrow (y, x)$

Let’s look at an example where we’ll reflect triangle ABC over the line $y = x$ using the coordinates.

$r_{y = x} \triangle{ABC}$

We know that the rule is the coordinates $(x, y)$ is going to switch to $(y, x)$ . We’ll simply switch them.

$A (3, 3) \rightarrow A^\prime (3, 3)$

$B (2, 1) \rightarrow B^\prime (1, 2)$

$C (6, 2) \rightarrow C^\prime (2, 6)$

Let’s graph this now. And draw a triangle.

Now, we have a reflection of triangle $ABC$ over the line $y = x$ to form the image of $A^\prime B^\prime C^\prime$ .