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

Reflecting a Shape in y = x Using Cartesian Coordinates

A shape can be reflected in the line y = x.

If point on a shape is reflected in the line y = x, the x-coordinate becomes the y-coordinate and the y-coordinate becomes the x-coordinate.

The image below shows a point on a shape being reflected in the line y = x:

  • The point A has Cartesian coordinates (2, 3).

  • The reflected point A' has Cartesian coordinates (3, 2).

    The x-coordinate of A (2) has become the y-coordinate of A'. The y-coordinate of A (3) has become the x-coordinate of A'.

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

Reflecting a shape in the line y = x using Cartesian coordinates is easy.

Question

Reflect the shape below in the line y = x.

Step-by-Step:

1

Find the Cartesian coordinates of each point on the shape.

Write the x-coordinates and y-coordinates of each point.

Point x-coordinate y-coordinate
A 1 6
B 4 5
C 1 4

2

Find the Cartesian coordinates of the reflected points.

Swap the x-coordinate and the y-coordinate of the original point.

Point x-coordinate y-coordinate Point x-coordinate y-coordinate
A 1 6 A' 6 1
B 4 5 B' 5 4
C 1 4 C' 4 1

3

Plot the reflected points and draw in the shape.

Answer:

We have reflected the shape in the line y = x.

A Formula to Reflect a Point in y = x Using Cartesian Coordinates

In general, we write Cartesian coordinates as:

x is the x-coordinate. y is the y-coordinate. x and y can taken any number.

The reflected point has Cartesian coordinates:

The image below shows a general Cartesian coordinate being reflected in the line y = x:

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The slider below shows another real example of how to reflect a shape in the line y = x using Cartesian coordinates.

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See Also

What is geometry? What are transformations? What are Cartesian coordinates? What is the x-coordinate? What is the y-coordinate?