Understanding Translation in Geometry
A translation in geometry refers to shifting a point or shape in a specific direction without changing its size or orientation. It is described using a column vector. The vector indicates how far to move in the x-direction (horizontal) and y-direction (vertical).
For example, a translation vector of ( \begin{pmatrix} 3 \ 2 \end{pmatrix} ) means to move 3 units right and 2 units up. To find the image of a point under a translation, simply add the vector to the original coordinates.
Example: If point A has coordinates (2, 4) and is translated using the vector ( \begin{pmatrix} 3 \ 2 \end{pmatrix} ):
- New coordinates = ( (2 + 3, 4 + 2) = (5, 6) ) Thus, the image of point A after translation is (5, 6).
For shapes, apply the same vector to each vertex to find the new positions.
Key points to remember
- Translation shifts points or shapes without rotation or resizing.
- A translation is described using a column vector.
- Add the translation vector to the original coordinates.
- Each vertex of a shape is translated individually.
- The direction and distance are defined by the vector.
Worked example
Translate point B (1, 2) using vector ( \begin{pmatrix} -4 \ 3 \end{pmatrix} ). New coordinates = ( (1 - 4, 2 + 3) = (-3, 5) ).