The Crossed Ladder Puzzle
 The
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 street.
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:
1/e+1/d=1/f
a2 = d2 + x2,
b2 = e2 + x2
Eliminating x gives you the following quartic equations:
e4 - 2fe3 + (f-e)2(a2-b2) = 0
d4 - 2fd3 + (f-d)2(b2-a2) = 0
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:
e4 - 70e3 + 3456e2 - 241920e +
4233600
d4 - 70d3 - 3456d2 + 241920d - 4233600
Synthetic division quickly gives e=60, d=84. Then the Pythagorean
Theorem gives you x=63. |
The Crossed Ladder Theorem
 |
The heart of the Crossed Ladder Puzzle is the fact that 1/e + 1/d = 1/f.
This is called the Crossed Ladder Theorem.
Proof:
By similar triangles,
AF'/AB = f/d, and
BF'/AB = f/e
The sum of AF'/AB and BF'/AB is 1, so
1 = f/d + f/e
Dividing by f, the result follows:
1/f = 1/d + 1/e
|
A Triangle Area Puzzle
|
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,
1/c + 1/f = 1/d + 1/e
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:
e = (5+10)/w
f = 10/w
d = (8+10)/w,
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. |
The Extended Crossed Ladder Theorem
A.k.a the "Crossed Ladders in a Triangle" theorem
|
1/c + 1/f = 1/d + 1/e
|
Proof:
A simple proof of the Extended Crossed Ladder Theorem works by drawing
two additional lines, representing buildings, which are parallel to the
four lines labeled c, d, e, and f. We extend the two ladders and two
sides of the triangle to meet these two new buildings.
We label the heights of the points where the extended ladders meet the
buildings as a and b. We label the heights of the points where the
extended triangle sides meet the buildings as x and y. Note x and y
encompass the entire heights of the buildings. In other words, part
of line x overlaps a, and part of line y overlaps b.
Now there are four crossed ladders, crossing at points C, D, E, and
F. From the standard Crossed Ladder Theorem, we get the following
identities:
1/c = 1/x + 1/y
1/f = 1/b + 1/a
1/d = 1/x + 1/b
1/e = 1/y + 1/a
By adding the first two and adding the second two, we get
1/c + 1/f = 1/x + 1/y + 1/a + 1/b
1/d + 1/e = 1/x + 1/y + 1/a + 1/b
So 1/c + 1/f = 1/d + 1/e, and the theorem is proved.
|
Final Remark
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.
Internet References
Mathworld: Crossed
Ladders Problem, Crossed
Ladders Theorem
Jimloy.com: The Crossed
Ladder Problem
Related pages in this website:
Summary of geometrical theorems
|
