traditional "crossed ladder" puzzle begins by telling you there are two
buildings, A and B, separated by a city street of width x. There are two
ladders, AD of length a and BE of length b, are positioned as shown. The
puzzle gives you f, a, and b and asks you to find x, the width of the
The solution depends on the "Crossed Ladder Theorem", which says 1/e + 1/d = 1/f. This, together with the Pythagorean Theorem, gives you the following equations:
Eliminating x gives you the following quartic equations:
The solutions to quartics are not easy, but numeric solutions aren't hard. Also, in typical contrived examples, the quartics can be easily factored. Jimloy.com's example is a case in point: a=105, b=87, f=35. These values give you the following quartics:
Synthetic division quickly gives e=60, d=84. Then the Pythagorean Theorem gives you x=63.
The heart of the Crossed Ladder Puzzle is the fact that 1/e + 1/d = 1/f. This is called the Crossed Ladder Theorem.
By similar triangles,
The sum of AF'/AB and BF'/AB is 1, so
Dividing by f, the result follows:
A triangle is divided into four regions by two straight lines. The areas of three of the regions are given. What is the area of the fourth region. (Note: diagram is not drawn to scale)
|The solution to the puzzle involves a variant of the
Crossed Ladder Theorem, which I will call the "Extended Crossed Ladder
Theorem", which states,
For our puzzle, the values of e, f, and d in the diagram to the left are clear from the areas in the diagram above:
where w is (1/2)(AB).
From the Extended Crossed Ladder Theorem, w/c + w/10 = w/18 + w/15, so c=45/w, which means the area of the entire triangle is 45. To find the indicated area, we subtract the other areas from 45, so the answer to the puzzle is 45 - 10 - 5 - 8 = 22.
A.k.a the "Crossed Ladders in a Triangle" theorem
1/c + 1/f = 1/d + 1/e
In the foregoing diagrams, the lines labeled a, x, b, y, c, d, e, and f all appear to be perpendicular to the base, AB. This isn't actually necessary to the validity of either of the Crossed Ladder Theorems, and in fact, I've taken care to avoid saying that. The only condition that is necessary for these theorems to be true is that the lines all be parallel to one another. The reason for this is due to the similar triangles that result from dropping perpendiculars from each of these same points -- the lengths of the perpendiculars are all in proportion to the corresponding lines.
Mathworld: Crossed Ladders Problem, Crossed Ladders Theorem
Jimloy.com: The Crossed Ladder Problem
Summary of geometrical theorems
The webmaster and author of this Math Help site is Graeme McRae.