When a question asks for a length or area inside a complex configuration of squares, circles or lines, pure Euclidean geometry can be slow. Instead, place the origin at a convenient corner or centre and assign (x,y) coordinates to the key vertices.
Trigger: An irregular length or a strange shaded area inside a square or along a line, with numerical side lengths given.
Instances:
(i) to find an area, subtract the areas of standard shapes (rectangles, right-angled triangles) whose vertices you just found;
(ii) to find a length, use Pythagoras on the coordinate differences;
(iii) for circle intersections, write down the equation x2+y2=r2 and find the intersection algebraically.
