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# Reflection in Geometry

Did you ever look in the mirror first thing in the morning and surprised yourself by how bad that fight with your pillow went last night, or maybe by how particularly good you look that morning? The truth is that mirrors don't lie, whatever is in front of them will be reflected without changing any of its features (whether we like it or not).

Let's start by defining what reflection is, in the context of Geometry.

## Definition of Reflection in Geometry

In Geometry, reflection is a transformation where each point in a shape is moved an equal distance across a given line. The line is called the line of reflection.

This type of transformation creates a mirror image of a shape, also known as a flip.

The original shape being reflected is called the pre-image, whilst the reflected shape is known as the reflected image. The reflected image has the same size and shape as the pre-image, only that this time it faces the opposite direction.

## Example of Reflection in Geometry

Let's take a look at an example to understand more clearly the different concepts involved in reflection.

Figure 1 shows a triangle shape at the right-hand side of the y-axis (pre-image), that has been reflected over the y-axis (line of reflection), creating a mirror image (reflected image).

Fig. 1. Reflection of a shape over the y-axis example

The steps that you need to follow to reflect a shape over a line are given later in this article. Read on if you want to know more!

## Real Life Examples of Reflection in Geometry

Let's think about where we can find reflections in our daily lives.

a) The most obvious example will be looking at yourself in the mirror, and seeing your own image reflected on it, facing you. Figure 2 shows a cute cat reflected in a mirror.

Fig. 2. Real life example of reflection - A cat reflected in a mirror

Whatever or whoever is in front of the mirror will be reflected on it.

b) Another example could be the reflection that you see in water. However, in this case, the reflected image can be slightly distorted in comparison to the original one. See Figure 3.

Fig. 3. Real life example of reflection - A tree reflected in water

c) You can also find reflections on things made out of glass, like shop windows, glass tables, etc. See Figure 4.

Fig. 4. Real life example of reflection - People reflected on glass

Now let's dive into the rules that you need to follow to perform reflections in Geometry.

## Reflection Rules in Geometry

Geometric shapes on the coordinate plane can be reflected over the x-axis, over the y-axis, or over a line in the form $$y = x$$ or $$y = -x$$. In the following sections, we will describe the rules that you need to follow in each case.

### Reflection over the x-axis

The rule for reflecting over the x-axis is shown in the table below.

 Type of Reflection Reflection Rule Rule Description Reflection over the x-axis $(x, y) \rightarrow (x, -y)$ The x-coordinates of the vertices that form part of the shape will remain the same.The y-coordinates of the vertices will change sign.

The steps to follow to perform a reflection over the x-axis are:

• Step 1: Following the reflection rule for this case, change the sign of the y-coordinates of each vertex of the shape, by multiplying them by $$-1$$. The new set of vertices will correspond to the vertices of the reflected image.

$(x, y) \rightarrow (x, -y)$

• Step 2: Plot the vertices of the original and reflected images on the coordinate plane.

• Step 3: Draw both shapes by joining their corresponding vertices together with straight lines.

Let's see this more clearly with an example.

A triangle has the following vertices $$A = (1, 3)$$, $$B = (1, 1)$$ and $$C = (3, 3)$$. Reflect it over the x-axis.

Step 1: Change the sign of the y-coordinates of each vertex of the original triangle, to obtain the vertices of the reflected image.

\begin{align}\textbf{Pre-image} &\rightarrow \textbf{Reflected image} \\ \x, y) &\rightarrow (x, -y) \\ \\A= (1, 3) &\rightarrow A' = (1, -3) \\ \\B = (1, 1) &\rightarrow B' = (1, -1) \\ \\C = (3, 3) &\rightarrow C' = (3, -3)\end{align}Steps 2 and 3: Plot the vertices of the original and reflected images on the coordinate plane, and draw both shapes. Fig. 5. Reflection over the x-axis example Notice that the distance between each vertex of the pre-image and the line of reflection (x-axis) is the same as the distance between their corresponding vertex on the reflected image and the line of of reflection. For example, the vertices \(B = (1, 1) and $$B' = (1, -1)$$ are both 1 unit away from the x-axis.

### Reflection over the y-axis

The rule for reflecting over the y-axis is as follows:

 Type of Reflection Reflection Rule Rule Description Reflection over the y-axis $(x, y) \rightarrow (-x, y)$ The x-coordinates of the vertices that form part of the shape will change sign.The y-coordinates of the vertices will remain the same.

The steps to follow to perform a reflection over the y-axis are as pretty much the same as the steps for reflection over the x-axis, but the difference is based of the on the change in the reflection rule. The steps in this case are as follows:

• Step 1: Following the reflection rule for this case, change the sign of the x-coordinates of each vertex of the shape, by multiplying them by $$-1$$. The new set of vertices will correspond to the vertices of the reflected image.

$(x, y) \rightarrow (-x, y)$

• Step 2: Plot the vertices of the original and reflected images on the coordinate plane.

• Step 3: Draw both shapes by joining their corresponding vertices together with straight lines.

Let's look at an example.

A square has the following vertices $$D = (1, 3)$$, $$E = (1, 1)$$, $$F = (3, 1)$$ and $$G = (3, 3)$$. Reflect it over the y-axis.

Step 1: Change the sign of the x-coordinates of each vertex of the original square, to obtain the vertices of the reflected image.

\begin{align}\textbf{Pre-image} &\rightarrow \textbf{Reflected image} \\ \x, y) &\rightarrow (-x, y) \\ \\D= (1, 3) &\rightarrow D' = (-1, 3) \\ \\E = (1, 1) &\rightarrow E' = (-1, 1) \\ \\F = (3, 1) &\rightarrow F' = (-3, 1) \\ \\G = (3, 3) &\rightarrow G' = (-3, 3)\end{align}Steps 2 and 3: Plot the vertices of the original and reflected images on the coordinate plane, and draw both shapes. Fig. 6. Reflection over the y-axis example ### Reflection over the lines y = x or y = -x The rules for reflecting over the lines \(y = x or $$y = -x$$ are shown in the table below:

 Type of Reflection Reflection rule Rule Description Reflection over the line $$y = x$$ $(x, y) \rightarrow (y, x)$ The x-coordinates and the y-coordinates of the vertices that form part of the shape swap places. Reflection over the line $$y = -x$$ $(x, y) \rightarrow (-y, -x)$ In this case, the x-coordinates and the y-coordinates besides swapping places, they also change sign.

The steps to follow to perform a reflection over the lines $$y = x$$ and $$y = -x$$ are as follows:

• Step 1: When reflecting over the line $$y = x$$, swap the places of the x-coordinates and the y-coordinates of the vertices of the original shape.

$(x, y) \rightarrow (y, x)$

When reflecting over the line $$y = -x$$, besides swaping the places of the x-coordinates and the y-coordinates of the vertices of the original shape, you also need to change their sign, by multiplying them by $$-1$$.

$(x, y) \rightarrow (-y, -x)$

The new set of vertices will correspond to the vertices of the reflected image.

• Step 2: Plot the vertices of the original and reflected images on the coordinate plane.

• Step 3: Draw both shapes by joining their corresponding vertices together with straight lines.

Here are a couple of examples to show you how these rules work. First let's perform a reflection over the line $$y = x$$.

A triangle has the following vertices $$A = (-2, 1)$$, $$B = (0, 3)$$ and $$C = (-4, 4)$$. Reflect it over the line $$y = x$$.

Step 1: The reflection is over the line $$y = x$$, therefore, you need to swap the places of the x-coordinates and the y-coordinates of the vertices of the original shape, to obtain the vertices of the reflected image.

\begin{align}\textbf{Pre-image} &\rightarrow \textbf{Reflected image} \\ \x, y) &\rightarrow (y, x) \\ \\A= (-2, 1) &\rightarrow A' = (1, -2) \\ \\B = (0, 3) &\rightarrow B' = (3, 0) \\ \\C = (-4, 4) &\rightarrow C' = (4, -4)\end{align}Steps 2 and 3: Plot the vertices of the original and reflected images on the coordinate plane, and draw both shapes. Fig. 7. Reflection over the line \(y = x example

Now let's see an example reflecting over the line $$y = -x$$.

A rectangle has the following vertices $$A = (1, 3)$$, $$B = (3, 1)$$, $$C = (4, 2)$$, and $$D = (2, 4)$$. Reflect it over the line $$y = -x$$.

Step 1: The reflection is over the line $$y = -x$$, therefore, you need to swap the places of the x-coordinates and the y-coordinates of the vertices of the original shape, and change their sign, to obtain the vertices of the reflected image.

\begin{align}\textbf{Pre-image} &\rightarrow \textbf{Reflected image} \\ \x, y) &\rightarrow (-y, -x) \\ \\A= (1, 3) &\rightarrow A' = (-3, -1) \\ \\B = (3, 1) &\rightarrow B' = (-1, -3) \\ \\C = (4, 2) &\rightarrow C' = (-2, -4) \\ \\D = (2, 4) &\rightarrow D' = (-4, -2)\end{align}Steps 2 and 3: Plot the vertices of the original and reflected images on the coordinate plane, and draw both shapes. Fig. 8. Reflection over the line \(y = -x example

## Reflection Formulas in Coordinate Geometry

Now that we have explored each reflection case separately, let's summarize the formulas of the rules that you need to keep in mind when reflecting shapes on the coordinate plane:

 Type of Reflection Reflection Rule Reflection over the x-axis $(x, y) \rightarrow (x, -y)$ Reflection over the y-axis $(x, y) \rightarrow (-x, y)$ Reflection over the line $$y = x$$ $(x, y) \rightarrow (y, x)$ Reflection over the line $$y = -x$$ $(x, y) \rightarrow (-y, -x)$

## Reflection in Geometry - Key takeaways

• In Geometry, reflection is a transformation where each point in a shape is moved an equal distance across a given line. The line is called the line of reflection.
• The original shape being reflected is called the pre-image, whilst the reflected shape is known as the reflected image.
• When reflecting a shape over the x-axis, change the sign of the y-coordinates of each vertex of the original shape, to obtain the vertices of the reflected image.
• When reflecting a shape over the y-axis, change the sign of the x-coordinates of each vertex of the original shape, to obtain the vertices of the reflected image.
• When reflecting a shape over the line $$y = x$$, swap the places of the x-coordinates and the y-coordinates of the vertices of the original shape, to obtain the vertices of the reflected image.
• When reflecting a shape over the line $$y = -x$$, swap the places of the x-coordinates and the y-coordinates of the vertices of the original shape, and change their sign, to obtain the vertices of the reflected image.

In Geometry, reflection is a transformation where each point in a shape is moved an equal distance across a given line. The line is called the line of reflection.

It depends on the type of reflection being performed, as each type of reflection follows a different rule. The rules to consider in each case are:

• Reflection over the x-axis → (x, y) when reflected becomes (x, -y).
• Reflection over the y-axis → (x, y) when reflected becomes (-x, y).
• Reflection over the line y = x →  (x, y) when reflected becomes (y, x).
• Reflection over the line y = -x →  (x, y) when reflected becomes (-y, -x).

A triangle with vertices A (-2, 1), B (1, 4), and C (3, 2) is reflected over the x-axis. In this case, we change the sign of the y-coordinates of each vertex of the original shape. Therefore, the vertices of the reflected triangle are A' (-2, -1), B' (1, -4), and C' (3, -2).

• Reflection over the x-axis → (x, y) when reflected becomes (x, -y).
• Reflection over the y-axis → (x, y) when reflected becomes (-x, y).
• Reflection over the line y = x →  (x, y) when reflected becomes (y, x).
• Reflection over the line y = -x →  (x, y) when reflected becomes (-y, -x).

The most obvious example will be looking at yourself in the mirror, and seeing your own image reflected on it, facing you. Other examples include reflections in water and on glass surfaces.

## Final Reflection in Geometry Quiz

Question

What is reflection in Geometry?

In Geometry, reflection is a transformation where each point in a shape is moved an equal distance across a given line. The line is called the line of reflection.

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Question

What is the original shape being reflected called?

Pre-image.

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Question

What is the reflected shape known as?

Reflected image.

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Question

The reflected image has the same size and shape as the pre-image.

True.

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Question

The reflected image faces the opposite direction of the pre-image.

True.

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Question

When reflecting a shape over the x-axis, what coordinate of each vertex of the original shape changes sign, to obtain the vertices of the reflected image?

y-coordinate.

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Question

When reflecting a shape over the y-axis, what coordinate of each vertex of the original shape changes sign, to obtain the vertices of the reflected image?

x-coordinate.

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Question

When reflecting a shape over the line $$y = x$$, what do you need to do to obtain the vertices of the reflected image?

The x-coordinates and the y-coordinates of the vertices that form part of the shape swap places.

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Question

When reflecting a shape over the line $$y = -x$$, what do you need to do to obtain the vertices of the reflected image?

The x-coordinates and the y-coordinates besides swapping places, they also change sign.

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Question

A triangle has the following vertices $$A = (5, 3)$$, $$B = (1, 2)$$, and $$C = (4, -1)$$. What will be the coordinates of the reflected image over the y-axis?

$$A = (-5, 3)$$, $$B = ( -1, 2)$$, and $$C = (-4,-1)$$.

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Question

A square has the following vertices $$A = (1, 1)$$, $$B = (3, 1)$$, $$C = (3, 3)$$, and $$D = (1, 3)$$. What will be the coordinates of the vertices of the reflected image over the x-axis?

$$A = (1, -1)$$, $$B = (3, -1)$$, $$C = (3, -3)$$, and $$D = (1, -3)$$.

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Question

A triangle has the following vertices $$A = (-3, -2)$$, $$B = (-2, 1)$$, and $$C = (-5, 2)$$. What are the coordinates of the vertices of the reflected image when reflected over the line $$y = -x$$?

$$A = (2, 3)$$, $$B = (-1, 2)$$, and $$C = (-2, 5)$$.

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Question

A triangle has the following vertices $$A = (-3, -2)$$, $$B = (-2, 1)$$, and $$C = (-5, 2)$$. What are the coordinates of the vertices of the reflected image when reflected over the line $$y = x$$?

$$A = (-2, -3)$$, $$B = (1, -2)$$, and $$C = (2, -5)$$.

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Question

A square has the following vertices $$A = (1, 1)$$, $$B = (3, 1)$$, $$C = (3, 3)$$, and $$D = (1, 3)$$. What will be the coordinates of the vertices of the reflected image over the line $$y = -x$$?

$$A = (-1, -1)$$, $$B = (-1, -3)$$, $$C = (-3, -3)$$, and $$D = (-3, -1)$$.

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