Even though it was now middle of winter, Jack hauled out his ladder and placed his ladder against the side of the house and began to climb. He had some bad luck when he reached the half-way point. The ground was a frozen sheet of ice and the base of his ladder slipped out and the top slid down the side of the house. Jack, clinging to the center rung, wound up moving from Point A on the side of his house to Point B on the ground.

Describe the path Jack traveled.

If the ladder is 1 unit long then at any point of its slide it looks like

                      |
                      |
(0,sqrt(1-x^2))|\
                      |  \
                      |    \
                      |      \
                      |        \
                      |______\____
                    (0,0)    (X,0)

The midpoint of the ladder has coordinates (x,y)=(X/2,(sqrt(1-X^2))/2) which satisfy x^2+y^2=1/4.  The midpoint of the ladder is thus on a circle with radius 1/2 centered at the origin.

Posted by Richard on 2004-07-27 15:00:39