Final answer:
The coordinates of point Q1 after a 180-degree clockwise rotation around point M are found by inverting its original coordinates, similar to points P1 and R1, due to the properties of rotation preserving distances and angles within the triangle.
Step-by-step explanation:
The question asks about the rotation of triangle PQR and the resulting coordinates of point Q1 after a 180-degree clockwise rotation around point M. It is assumed that we already know the new coordinates for points P1 and R1 after the rotation. Since the rotation is 180 degrees, any point will be mapped to a point directly opposite the center of rotation at the same distance. This invariance in distance is captured by the concept that the distance of a point to the origin (or a rotation center) is preserved during rotation, as stated in [x² + y²]. The relation for the rotated coordinates, x' and y', according to a general rotation by angle θ (theta) is given by: x' = x cos θ + y sin θ and y' = -x sin θ + y cos θ. When θ is 180 degrees or π radians, cos(π) = -1, and sin(π) = 0. Therefore, the formulas simplify to x' = -x and y' = -y. Given the new coordinates of P1 and R1, the new coordinates of Q1 can be deduced by maintaining the geometric relationships within the triangle post-rotation.