Reflection in Geometry

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!

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

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

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

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

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