Given a wall perpendicular to the ground and ladder leaning against it. What is the path of the midpoint of the ladder as the top slide up or down the wall. How do you know?

The object created by the wall and the ladder is a right triangle. The ladder has a fixed length, therefore the length of the midpoint from the orgin is static. As shown by the blue segments, by the median of a right triangle theorem, the length of the median (CM) is equal length to the two segments of the hypoteneuse. That is CM=BM=MA. Since CM is constant, the path traced by the midpoint of the ladder is a quarter circle where the orgin is the center and is the locus of the distance CM away from the orgin is the path traced by the midpoint.