The midpoint of a sliding ladder

Question

A ladder of length L stands upright against a vertical wall on a horizontal floor. Its foot slides away from the wall, the top sliding down, until the ladder lies flat on the floor.

What curve does the midpoint of the ladder trace out? Prove it. Then say what a point one quarter of the way along the ladder traces.

Accepted Answer

A quarter circle of radius L/2 centred at the corner where the wall meets the floor. Why. At any moment the ladder is the hypotenuse of a right-angled triangle formed with the wall and the floor, with the right angle at the corner. In a right triangle, the midpoint of the hypotenuse is the centre of the circle through all three corners (the hypotenuse is a diameter), so it is equally distant from all three corners. In particular its distance from the corner is always half the hypotenuse, L/2. A point that stays at fixed distance L/2 from the corner moves on a circle of radius L/2 centred ther…

The midpoint of a sliding ladder

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