Final answer:
The student's question pertains to solving advanced plane geometry and similarity problems that involve vector analysis, coordinate conversions, and applications of geometric principles in real-life scenarios like navigation and astronomy.
Step-by-step explanation:
The student's question relates to the application of plane geometry and the concept of similarity to solve problems in various contexts including navigation, vector analysis, and real-world scenarios. One problem involves finding component distances of a displacement in two different orientations, another requires the conversion of coordinates from rectangular to polar format, and another addresses the process of finding polar coordinates of given points.
Vectors and their resultant through geometric constructions, and an understanding of congruent triangles in astronomical observations, also form part of the problems presented.
These problems are solved by applying mathematical principles and understanding how geometric shapes and coordinates relate to real-world measurements. For instance, vector components and resultants are crucial in navigation and physics, while coordinate conversions are fundamental in many areas of engineering and computer science.