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How to Reflect a Shape In the Line y = x (Mathematics Lesson)

How to Reflect a Shape in the Line y = x

To reflect a shape in the line y = x, for each vertex (corner) of the shape:



Step 1: Find the diagonal distance to the line y = x.

Step 2: Draw the reflected vertex the same diagonal distance on the other side of the line y = x.

When all vertices have been reflected, draw the reflected shape.

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

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



  • For point A:

    Step 1: Find the diagonal distance to the line y = x.
    Point A is 2½ diagonals from the line y = x.



    Step 2: Draw the reflected vertex the same distance on the other side of the line y = x.


  • For point B:

    Step 1: Find the diagonal distance to the line y = x.
    Point B is ½ a diagonal from the line y = x.



    Step 2: Draw the reflected vertex the same distance on the other side of the line y = x.


  • For point C:

    Step 1: Find the diagonal distance to the line y = x.
    Point C is 1½ diagonals from the line y = x.



    Step 2: Draw the reflected vertex the same distance on the other side of the line y = x.

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



This is the shape reflected in the line y = x.

How to Reflect a Shape in the Line y = x Using Cartesian Co-ordinates

In the above example, the shape has been reflected by reading off visually the distance of points from the line y = x.

An alternative method transforms the Cartesian co-ordinates of each point (x, y) to its reflected point (y, x).



Read more about how to reflect a shape in the line y = x using Cartesian co-ordinates
Curriculum
Geometry Lessons
Interactive Test
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Note
WHAT IS A REFLECTION?

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

By reflection, think of the reflection you would see in a mirror or in water. The orginal shape is called the object, the reflected shape is called the image:



It is as if the shape has been flipped.

A reflection is a type of transformation.

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.



Top Tip
YOU DON'T HAVE TO MEASURE DIAGONALLY TO THE LINE

In the real example, the diagonal distance of each vertex to the line is measured.

The reflected vertex is drawn the same diagonal distance on the other side of the line:



However, the measurement does not have to be diagonal. You can measure horizontally or vertically:

  • Measure the vertical distance to the line y = x:



    The reflected vertex is the same horizontal distance from y = x:

or

  • Measure the horizontal distance to the line y = x:



    The reflected vertex is the same vertical distance from y = x:

So, a vertex of a shape can be reflected 3 ways:

  • diagonally
  • vertical then horizontal
  • horizontal then vertical