Let x and y be positive real numbers. Four points are defined as follows:
A(x,y), B(−x,y), C(−x,−y) and D(x,−y).
(i) For each point A,B,C and D, state the quadrant in which it lies or whether it lies on an axis. Give reasons based on the signs of its coordinates.
(ii) Compare the abscissa and ordinate of each pair of opposite points (for example, A with C and B with D). Explain how the change in sign creates symmetry about the axes.
(iii) If the four points are joined in order A→B→C→D→A, name the geometric figure obtained. Describe which quadrants and axes this figure touches or crosses, supporting your answer with coordinate reasoning.