.
A ladder that is 6m long rests with one end on the horizontal ground and its other end against a vertical wall. Considering the ground and the wall as the x and y axes respectively, find the locus of the possible midpoints of the ladder.
----------------------------------------------------------------------
Answer
The locus is the arc of a circle, namely, a 90° arc.
The center of the circle is at the base of the wall.
The radius of the circle is 3m, which is half of the length of the ladder.
Proof
Let us connect the midpoint of the ladder with the base of the wall (I mean: let us draw this imaginary line . . . ).
This line is the median in our right-angled triangle.
It is well known fact that in an right-angled triangle the median drawn to the hypotenuse has the length
half of the hypotenuse (see the lesson Median drawn to the hypotenuse of a right triangle in this site).
Thus the midpoint of the ladder is always at the distance half of the ladder length from the base of the wall.
So the locus is the arc of the circle.
To estimate the angle of the arc, consider two extreme positions of the ladder: horizontal and vertical.