Crossed Ladders Problem


Given two crossed ladders resting against two buildings, what is the distance between the buildings? Let the height at which they cross be h and the lengths of the ladders l_1 and l_2. The height at which l_2 touches the building h_2 is then obtained by simultaneously solving the equations




the latter of which follows either immediately from the crossed ladders theorem or from similar triangles with d_1=dh/h_2, d_2=dh/h_1, and d=d_1+d_2. Eliminating d gives the equations


These quartic equations can be solved for h_1 and h_2 given known values of h, l_1, and l_2.

There are solutions in which not only l_1, l_2, h_1, h_2, and h are all integers, but so are d_1, and d_2. One example is (l_1,l_2,h_1,h_2,h,d_1,d_2)=(119,70,105,42,30,40,16).


The problem can also be generalized to the situation in which the ends of the ladders are not pinned against the buildings, but propped fixed distances d_1 and d_2 away.

See also

Crossed Ladders Theorem

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Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 62-64, 1979.

Referenced on Wolfram|Alpha

Crossed Ladders Problem

Cite this as:

Weisstein, Eric W. "Crossed Ladders Problem." From MathWorld--A Wolfram Web Resource.

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