To translate triangle ABC to triangle A''B''C'', the rule is to sum the two translation vectors responsible for the consecutive translations.
This final translation vector is the vector addition of the individual vectors applied to the original triangle to achieve the final position.
The student wants to know the rule for translating triangle ABC to triangle A''B''C'' using vectors.
Since a translation can be described using a vector, which tells us how to move each point of a shape to its new location, the overall translation from △ABC to △A''B''C'' is the sum of the two translation vectors used to move △ABC to △A'B'C', and then △A'B'C' to △A''B''C''.
Based on the properties of vector addition, we know that the combination of two translations can be described by the sum of both translation vectors.
This is due to the commutative and distributive laws of vector operations.
Vector addition follows the parallelogram rule, wherein two vectors originating from the same point will form a parallelogram, and their sum is represented by the diagonal of that parallelogram.
If the first translation vector is v1 and the second translation vector is v2, then the rule to translate △ABC to △A''B''C'' is simply the sum of these two vectors, v1 + v2.