Translation Matrices

Translation Matrices

How to use a translation matrix to find the image under translation: the matrix, example, and its solution.

Matrix

For the translation (x, y) -> (x + a, y + b), the translation matrix is [a / b]. To find the coordinates of the image, add the translation matrix and the vertex matrix.

For the translation (x, y) → (x + a, y + b),
the translation matrix is [a / b].

To find the coordinates of the image,
add the vertex matrix and the translation matrix:
[x / y] + [a / b] = [x + a / y + b].

Example

Triangle ABC has vertices A(1, 2), B(5, 4), and C(3, 1). Its image is under the translation (x, y) -> (x + 7, y + 5). Find the coordinates of the vertices of the image.

Previously, you've solved this example.

Translation of a point

Let's solve the same example
by using the tranlation matrix.

Add the vertex matrix and the translation matrix:
[-2 2 0 / 1 3 -1] + [7 7 7 / 5 5 5] = [5 9 7 / 6 8 4].

Then the vertices of the image are
A'(5, 6), B'(9, 8), C'(7, 4).

You can see that
by adding the translation matrix, [7 7 7 / 5 5 5],
you can find the vertices of the image,
which is under the translation (x, y) → (x + 7, y + 5).