If the transformation preserve the congruents, so the distance between two points remains equal after the transformation, this is:
[tex]\begin{gathered} (x,y)\to(x^{\prime},y^{\prime}),\text{ preserve the congruents so:} \\ \text{If we have P}_1=(x_{1,}y_1)andP_2=(x_2,y_2)\text{ } \\ \text{And P'}_1^{}=(x^{\prime}_1,y^{\prime}_1),P^{\prime}_2=(x^{\prime}_2,y^{\prime}_2)\text{ are the points after the transformation, so:} \\ \sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}=\sqrt[]{(x^{\prime}_2-x^{\prime}_1)^2+(y^{\prime}_2-y^{\prime}_1)^2} \end{gathered}[/tex]
Let
