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