# What Is Reflection in Math? Definition, Examples & How-to

Aug 1, 2024 | Ankeny

Whether you are just starting to explore reflections in math or need to brush up on your geometry, this simple, middle-school-friendly guide is for you.

## What Is Reflection?

Reflection in mathematics is a geometric transformation where a shape or object is flipped across a line, known as the line of reflection. This results in a mirror image that is the same size and shape as the original but appears flipped or mirrored.

You encounter the concept of reflection every time you look in a mirror.

Try it now:

Step in front of a mirror and raise your right hand.

What hand is your mirror image raising?

If you raise your right hand in front of a mirror, your reflection, i.e. your mirror image, raises its left hand.

### Transformations in Geometry

Reflection is one of the 4 types of transformations in geometry .

Other types of geometrical transformations are:

• Translation: Moving a shape without rotating or flipping it. It's like sliding the shape in a particular direction.
• Rotation: Turning a shape around a fixed point. It’s like turning a key in a lock.
• Dilation: Resizing a shape by enlarging or shrinking it evenly. It's like stretching or squeezing the shape while keeping its proportions intact, like inflating or deflating a balloon.

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### Properties of Reflection

To truly understand what reflection is and to distinguish it from other geometrical transformations, we need to know its properties.

Reflection has 5 key properties:

• Shape: The shape of the original figure and its mirror image are identical.
• Size: The size of the original figure and its mirror image are the same.
• Orientation: The orientation of the original figure and its mirror image are opposite.
• Distance: The distance between any point on the original figure and the line of reflection is the same as the distance between the corresponding point on the mirror image and the line of reflection.
• Angle: Angles between intersecting lines are the same in both the original figure and its mirror image.

If any of these five properties are missing, the geometrical transformation is either not a reflection at all or is a combination of reflection and other transformations.

### Important Terms in Reflection

Let’s go over some language we use to talk about reflections.

We use these terms to explain how reflection works:

• Line of Reflection: The imaginary line across which a shape is reflected to create its mirror image. It is also known as the axis of reflection or mirror line.
• Mirror Image: The image formed when you reflect a shape across the line of reflection. It is identical to the original shape but appears reversed.
• Congruent Figures: Two figures that have the same size and shape. In reflection, the original shape and its mirror image are always congruent.
• Symmetry: The property of a shape or object that remains unchanged when reflected across a line, known as a line of symmetry.

### How to Do Reflections on the Coordinate Plane

To reflect a point or figure on the coordinate plane, we sometimes use the X-axis or Y-axis as the line of reflection.

Let’s see how each type of reflection works.

### How to DoReflection Over the X-axis

Reflection over the x-axis is a transformation where each point in a shape or a graph is flipped across the x-axis.

If you have a point (x, y), reflecting it over the x-axis will give you the point (x, -y).

In other words, the x-coordinate (how far left or right the point is) stays the same, but the y-coordinate (how far up or down the point is) becomes its opposite.

For example, if you have the point (2, 3), reflecting it over the x-axis would give you (2, -3), because the x-coordinate remains 2, but the y-coordinate changes from 3 to -3, flipping it across the x-axis.

### How to Do Reflection Over the Y-axis

In reflection over the Y-axis, each point in a shape or graph is mirrored horizontally along the Y-axis.

When you have a point (x, y), reflecting it over the Y-axis gives you the point (-x, y). To put it simply, the y-coordinate remains unchanged, but the x-coordinate changes to its negative value.

For example, let's take the point (4, -5). Reflecting it over the Y-axis results in (-4, -5).

Notice that while the y-coordinate remains -5, the x-coordinate changes from 4 to -4 as it mirrors along the Y-axis.

Here’s a visual representation of reflection over the Y-axis.

### How to Do Reflection Over the Line y = x

The line y = x represents all the points where the y-coordinate is equal to the x-coordinate.

It's a diagonal line that passes from the bottom left corner to the top right corner through the origin at a 45-degree angle.

Reflection over the y = x line means flipping a point or shape across this diagonal line.

During this reflection, the x-coordinate of each point becomes its y-coordinate, and the y-coordinate becomes its x-coordinate.

For example, if you have a point (x, y), its reflection over the y = x line would be (y, x).

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