1. How to construct the data of an isometry from the image of a triangle
Given two triangles ABC and A'B'C', we say that a transformation T takes ABC to A'B'C' if T(A) = A' and T(B) = B', and T(C) = C'. If ABC and A'B'C' are congruent triangles, we know from a fundamental theorem on isometries that there is exactly one isometry T that takes ABC to A'B'C'.
Note: This statement takes into account the order of the vertices. If T takes ABC to A'B'C', then it does not take ABC to C'A'B'.
However, in specific cases, one needs to determine from the two triangles exactly what isometry is T. What type of isometry is it and what is its defining data.
Construction Method 1: (Use the proof of the fundamental theorem)
