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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:
ry=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)

Reflection Over the Line y=x

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.