# 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 and changing the sign of both coordinates.
(-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 (-6,3)
B (-3,2)
C (-6,1)

Step 2: Find the Cartesian coordinate of the reflected vertex by swapping the x coordinate and the y coordinate around and changing the sign of both coordinates.

Vertex of Object Coordinate Vertex of Image Coordinate
A (-6,3) A' (-3,6)
B (-3,2) B' (-2,3)
C (-6,1) C' (-1,6)

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 co-ordinate 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 top left to bottom 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.