Question

A ladder of unit length leans against a wall (y-axis) as its bottom slides

along the ground (x-axis). As the ladder slides from vertical to horizontal,

one end remains in contact with the wall while the other end remains in

contact with the ground.

Find :

(1) Derive an algebraic equation describing the trajectory traced out by

the midpoint of the ladder. What is the shape of the trajectory?

(2) Derive an algebraic equation for the envelope bounded by the ladder.

Answer 1

This theorem would lead quickly to the picture in Figure 1.
That is, as the ladder moves, the legs of the right triangle change but the hypotenuse (the ladder) remains the same length and the distance of its midpoint from
the center of the circle will stay the same. The locus of the midpoint then lies
along the arc of a circle with a radius determined by the distance of the midpoint of
the hypotenuse from the right-angle vertex.

Fig.1.Diagram illustrating that the midpoint of the hypotenuse of a right
triangle is equidistant from the three vertices.