February 21, 2019

## Geometry: Translations, Reflections, Rotations, and Dilations

One unit covered in Geometry deals with the concept of translations, reflections, rotations, and dilations.  This article will serve to summarize some the major points for students studying these topics.

Prerequisite:  The student needs to come into the lesson with some basic understanding of matrices.  Given a shape with points on a coordinate plane they need to be able to write those in matrix form.  It is very simple actually, you take the x-coordinates and make them the first row of your matrix and take the y-coordinates and make them the second row of your matrix.

Example: A quadrilateral with points (-4,-3), (-1,0), (1,-3), and (-3, -5) would be written as the following matrix:

$$\left[ \begin{array}{ c c c c } -4 & -1 & 1 & -3\\ -3 & 0 & -3 & -5 \end{array} \right]$$

Students also need to know the identity matrix when multiplied by a matrix gives back the original matrix.  It is like multiplying a number times 1.  The identity matrix is:

$$\left[ \begin{array}{ c c } 1 & 0 \\ 0 & 1 \end{array} \right]$$

Translations: Translations simply slide your figure around.  It is the easiest to work with since it just involves adding a value to the x-coordinates and a value to the y-coordinates.

Example: Translate the example matrix above by moving it to the RIGHT four and DOWN 1.  This would mean we just add 4 to the top numbers and subtraction 1 from the bottom numbers:

$$\left[ \begin{array}{ c c c c } -4 & -1 & 1 & -3\\ -3 & 0 & -3 & -5 \end{array} \right] + \left[ \begin{array}{ c c c c } 4 & 4 & 4 & 4\\ -1 & -1 & -1 & -1 \end{array} \right]= \left[ \begin{array}{ c c c c } 0 & 2 & 5 & 1\\ -4 & -1 & -4 & -6 \end{array} \right]$$

Dilations: Dilations make an object bigger or smaller.  If the dilation is a number bigger than 1, the object will increase in size; if it is less than 1, it will get smaller.  Dilations require you multiple each number in the given matrix by the dilation value.

Example: Dilate the given quadrilateral by 3.

3 * $$\left[ \begin{array}{ c c c c } -4 & -1 & 1 & -3\\ -3 & 0 & -3 & -5 \end{array} \right] = \left[ \begin{array}{ c c c c } -12 & -3 & 3 & -9\\ -9 & 0 & -9 & -15 \end{array} \right]$$

The “slightly” harder problems involve ROTATION and REFLECTION.

These simple “adjust” the coordinates according to a specific matrix.  Let’s look at some different matrices and see what they do:

Identity: Doesn’t change the value of the matrix.

$$\left[ \begin{array}{ c c } 1 & 0 \\ 0 & 1 \end{array} \right]$$

Matrix 1: Notice this looks just like the identity matrix except it has negative 1’s rather than positive ones.  This means that it will change each sign to its opposite in the matrix.

$$\left[ \begin{array}{ c c } -1 & 0 \\ 0 & -1 \end{array} \right]$$

Matrix 2: This matrix looks like the identity but has a negative only in the top 1.  This means only the top row will change to their opposite signs but the bottom row will stay the same.

$$\left[ \begin{array}{ c c } -1 & 0 \\ 0 & 1 \end{array} \right]$$

Matrix 3: Matrix 3 is similar to Matrix 2 but the negative is on the bottom instead of the top.  This means the bottom row will change to its opposite sign and the top row stays the same.

$$\left[ \begin{array}{ c c } 1 & 0 \\ 0 & -1 \end{array} \right]$$

Matrix 4: This matrix is a little different from the identity.  The 1’s and the 0’s have changed places.  When this happens, the whole row changes places.  Since both are positive, the numbers keep their original signs.

$$\left[ \begin{array}{ c c } 0 & 1 \\ 1 & 0 \end{array} \right]$$

Matrix 5: Can you guess what happens in this matrix?

$$\left[ \begin{array}{ c c } 0 & -1 \\ 1 & 0 \end{array} \right]$$

$$\left[ \begin{array}{ c c } 0 & 1 \\ -1 & 0 \end{array} \right]$$

Matrix 7: And this one?

$$\left[ \begin{array}{ c c } 0 & -1 \\ -1 & 0 \end{array} \right]$$

Matrix 5 – the rows switch places and the top row has opposite signs.

Matrix 6 – the rows switch place and the bottom row has opposite signs.

Matrix 7 – the rows switch places and both rows also change signs.

Each of these matrices are multiplied times the matrix defined by the shape in the problem  Note that the “identity” type matrix always comes first, then the other matrix so that the dimensions match for multiplying.

Here is a summary of when to use each matrix:

Reflection over y = x: use matrix

$$\left[ \begin{array}{ c c } 0 & 1 \\ 1 & 0 \end{array} \right]$$

Reflection over x-axis: use matrix

$$\left[ \begin{array}{ c c } 1 & 0 \\ 0 & -1 \end{array} \right]$$

Reflection over y = -x: use matrix

$$\left[ \begin{array}{ c c } 0 & -1 \\ -1 & 0 \end{array} \right]$$

Reflection over y-axis: use matrix

$$\left[ \begin{array}{ c c } -1 & 0 \\ 0 & 1 \end{array} \right]$$

Rotation of 90 degrees: use matrix

$$\left[ \begin{array}{ c c } 0 & -1 \\ 1 & 0 \end{array} \right]$$

Rotation of 180 degrees (same as rotation over Ho): use matrix

$$\left[ \begin{array}{ c c } -1 & 0 \\ 0 & -1 \end{array} \right]$$

Rotation of 270 degrees: use matrix

$$\left[ \begin{array}{ c c } 0 & 1 \\ -1 & 0 \end{array} \right]$$

Example: Given A(2,5) and B(1, -2) and C(-2,3).

Find a rotation of 270 degrees:

$$\left[ \begin{array}{ c c } 0 & 1 \\ -1 & 0 \end{array} \right] * \left[ \begin{array}{ c c c } -2 & 1 & -2 \\ 5 & -2 & 3 \end{array} \right] = \left[ \begin{array}{ c c c } 5 & -2 & 3\\ -2 & -1 & 2 \end{array} \right]$$

Note:  The two rows switched, then the bottom row switches signs.

Find a reflection over the y-axis:

$$\left[ \begin{array}{ c c } -1 & 0 \\ 0 & 1 \end{array} \right] * \left[ \begin{array}{ c c c } 2 & 1 & -2 \\ 5 & -2 & 3 \end{array} \right] = \left[ \begin{array}{ c c c } -2 & -1 & 2\\ 5 & -2 & 3 \end{array} \right]$$

Note: The rows did not switch, but the signs on the top row changed to their opposites.

To summarize:

Dilations: Multiply matrix through by the amount of the dilation.

Translation:  Adjust each x by the change in the x-axis of the translation and adjust each y by the change in the y-axis  of the translation.

Reflections and Rotations:  Find the corresponding matrix for each reflection or rotation and multiply the matrix by the correct “identity-type” matrix listed above.

Happy Transformations!