Final answer:
The truth of Triangle PQR being congruent to Triangle TSR cannot be determined without additional information. It is true that the Pythagorean theorem can calculate the length of a resultant vector from two right-angle vectors and that a vector can form a right triangle with its components.
Step-by-step explanation:
To answer the initial student's question regarding Line PQ being parallel to Line ST and R being the midpoint of Line PT, we cannot definitively prove that Triangle PQR is congruent to Triangle TSR without further information. Therefore, this statement could be either true or false depending on additional given details which are not provided in the question.
Regarding the use of the Pythagorean theorem for calculating the resultant vector when adding two vectors at right angles to each other, the statement is true. The theorem applies perfectly in this scenario, as the vectors form the legs of a right-angle triangle with the resultant vector being the hypotenuse. The lengths of the vectors (legs of the triangle) can be squared and added together to find the square of the length of the resultant vector (the hypotenuse).
It is also true that a vector can form the shape of a right-angle triangle with its x and y components, as these components act as the legs of the triangle, with the vector itself being the hypotenuse.
Regarding distance invariance under rotation of coordinate system, the distance between two points remains constant regardless of the rotation of the coordinate system. This is a fundamental attribute of Euclidean space and follows from the definition of distance using the Pythagorean theorem. The squared distance would be calculated as x2 + y2, and this value does not change with rotation since it's based on the fixed distance between points, not their specific coordinates in any given rotation of the axes.