Final answer:
Option a) rotation, translation, reflection is the correct set of transformations that will always produce a congruent triangle because all these transformations preserve the size and shape of geometric figures.
Step-by-step explanation:
The question is asking which set of transformations will always produce a congruent triangle. Congruence in geometry means that all corresponding sides and angles are identical in measure, but the figure may be rotated, reflected, or translated from its original position. To answer the question:
a) rotation, translation, reflection: These transformations preserve the size and shape of the triangle, resulting in a congruent figure. Rotation moves the triangle around a fixed point, translation slides the triangle without rotating it, and reflection flips the triangle over a line without changing its size or shape.
Regarding the rotation, the coordinates after rotation in terms of the original coordinates can be expressed as:
x' = x cos q + y sin q
y' = -x sin q + y cos q
We see that the parameters o and p are meant to represent the same angle q, which is a typo in the provided information. Rotation about a point follows these equations and maintains the lengths of sides and the measures of angles, thereby keeping the triangle congruent to the original.
Options b), c), and d) all include dilation, which changes the size of the triangle, thus they do not always produce a congruent triangle.