How to Reflect a Shape in the Line y = x Using Cartesian Coordinates

To reflect a shape in the line y = x using Cartesian coordinates, if a point on the shape has coordinates (x,y), the reflected point has coordinates (y,x).

Method of How to Reflect a Point in Line y = x Using Cartesian Coordinates

For each vertex (corner) of the shape:

Step 1: Find the Cartesian coordinate of the vertex.
(x,y)

Step 2: Find the Cartesian coordinate of the reflected vertex by swapping the x-coordinate and the y-coordinate around.
(y,x)

A Real Example of How to Reflect a Shape in the Line y = x Using Cartesian Coordinates

Question: Reflect the shape below in the line y = x.

Step 1: Find the Cartesian coordinates of each vertex.

Vertex of Object Coordinate
A (1,6)
B (4,5)
C (1,4)

Step 2: Find the Cartesian coordinate of the reflected vertex by swapping the x-coordinate and the y-coordinate around.

Vertex of Object Coordinate Vertex of Image Coordinate
A (1,6) A' (6,1)
B (4,5) B' (5,4)
C (1,4) C' (4,1)

With all the vertices reflected, we can draw in the reflected shape:

This is the shape reflected in the line y = x.
Geometry Lessons
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Note
FINDING THE CARTESIAN COORDINATES OF A POINT

• The coordinate on the left is the x-coordinate.

It describes how far along the x-axis, or how far across, the point is.

• The coordinate on the right is the y-coordinate.

It describes how far up the y-axis, or how far up, the point is.
For example, the point below is 2 along the x-axis and 3 up the y-axis. Therefore its Cartesian coordinates are (2,3)

WHAT IS THE LINE y = x

y = x is a linear equation.

y = x represents a diagonal line that goes through the origin. It goes from bottom left to top right.

It goes halfway between the x-axis and the y-axis. y = x is at 45° to both axes.

WHAT IS A REFLECTION?

A reflection transforms a shape so that it becomes a mirror image of itself.

The orginal shape is called the object, the reflected shape is called the image.

A reflection is a type of transformation.