# Translation

A Translation \(\alpha\) (also known as "slide") is a bijective mapping from \(\mathbf{R}^{2}\) to \(\mathbf{R}^{2} \) that sends point \((x,y)\) to \((x',y')\) such that the \(\alpha\) has equations \(\begin{cases} x' = x+a \\ y'=y+b \end{cases}\), for some \(a, b \in \mathbf{R} \). In other words, \(\alpha((x,y))=(x',y')\). Most of the times, \( \alpha \) (or any lower case greek letter) is used to represent any transformations (i.e., halftturns, reflection, rotation, dilation, etc.). We shall use the greek letter \(\tau\) for translations.

## Examples

For example, let \(A=(c,d)\) and \(B=(e,f) \) be points in \(\mathbf{R}^{2}\). Then there are unique numbers \(a\) and \(b\) such that \(e=c+a\) and \(f=d+b\). Thus, the unique translation that takes \(A\) to \(B\) has equations \(x'=x+(e-c)\) and \(y'=y+(f-d)\). We shall denote this unique translation by \(\tau_{A,B}\).

A simple illustration would be this,

Example 1. Given \(A=(5,6)\) and \(B=(7,10)\). Find \(\tau_{A, B}((1,2))=?\).

\(Answer\). \(\tau_{A,B}\) has equations \(\begin{cases} x'=x+(7-5) \\ y'=y+(10-6) \end{cases}\). Hence, \(\tau_{A,B}((1,2))=(3,6)\).

Example 2. Given \(A=(1,2)\), \(B=(3,4)\), and \(C=(5,6)\). Find \(\tau_{A, B}\tau_{B,C}\tau_{A,C}((1,2))=?\).

\(Answer\).\((9,10)\)

Example 3. Find the equations for \(\tau_{A,B}\tau_{B,A}\).

\(Answer\).\(x'=x\) and \(y'=y\).

Let \(\tau_{A, B}\) represent a linear translation in the \(x\)-\(y\) plane from point \(A\) to point \(B\).

Given, the points:

- \(A=(21,42)\)
- \(B=(32,43)\)
- \(C=(-12,6)\)
- \(D=(-112,206)\)

If \(\tau_{A, B}\tau_{B,C}\tau_{C,D}\tau_{D,A}(2,8)=(a,b)\), what is \(a+b\)?