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Crossed Ladders Problem


CrossedLaddersProblem

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

l_1^2=h_1^2+d^2
(1)
l_2^2=h_2^2+d^2
(2)

and

 1/h=1/(h_1)+1/(h_2),
(3)

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

h_1^4-2hh_1^3+(h-h_1)^2(l_2^2-l_1^2)=0
(4)
h_2^4-2hh_2^3+(h-h_2)^2(l_1^2-l_2^2)=0.
(5)

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).

CrossedLaddersProblem2

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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References

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. https://mathworld.wolfram.com/CrossedLaddersProblem.html

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