According to his doctoral dissertation, Christopher Powell of the Maya Exploration Center observed modern Maya using the geometric fact that a rhombus with equal diagonals must be a square to make sure the corners were square when laying out a building. This fact allows one to easily check whether or not a rhombus is a square by checking if the diagonals are equal with a cord or piece of rope. More generally, a parallelogram with equal diagonals must be a rectangle.
In a lecture during the 2011 MAA Study Tour, Powell explained another clever way that the Maya used a cord to form right angles in laying out a square that he learned from a master builder who had learned it while a shaman apprentice. The method uses a knotted cord and properties of equilateral triangles. The cord has eight knots on it, dividing the cord into seven equal segments with a knot at each end. There are loops at each of the knots for staking to the ground. Since the knots are evenly spaced, when knots 1 and 4 are held together and the cord pulled taut, an equilateral triangle with interior angles of 60° is formed. Then knot 6 is joined with knot 3 and the cord pulled tight resulting in another equilateral triangle formed by knots 4, 5 and 6. Finally knot 8 is joined with knot 5 forming a third equilateral triangle (and all together, half of a hexagon). If one runs a ray (or rope) from knot 1 through knot 2 and another ray from knot 1 through knot 7, the resulting angle is a right (60° + 30°) angle. The animation below (created by the first author) illustrates how the right angle is formed.
The first author also created the following video using student volunteers to demonstrate the Maya way of forming a right angle. See Activity 3 for detailed instructions for recreating this activity with your students.