Does congruence guarantee length conversion? Suppose that a linear transformation $M:R^2 \rightarrow R^2$ maps a triangle $ABC$ to a congruent triangle $A'B'C'$
($\{A, B, O\}, \{B, C, O\},\{C, A, O\}$ are not colinear, and $A,B,C\neq O$)
Is it true that the linear transformation always conserves length?
Or can there possibly be a counterexample?
(I'm talking about transformations on $R^2$ that can be represented as a 2 by 2 matrix)
Thanks in advance

By definition, all congruent triangles have the same size. So you can map the sides $AB$, $BC$, $CA$ to $A'B'$, $B'C'$ and $C'A'$, respectively. Now let's consider if this mapping is an isometry. Clearly, $|AB|=|A'B'|$, as we have defined by congruency. So the vector $AB$ is mapped to another vector of the same length. The same happens to the vector $BC$ mapped to $B'C'$. These vectors however are linearly independend, as you defined. Now you use the fact that a linear mapping is a isometry iff the mapping of each basis vector keeps the original length constant.
