There are two Russian Army motorcyclists. In the section from a map given in our illustration below we are shown three long straight roads, forming a right-angled triangle at C.
The General asked the two men how far it was from A to B. Pipipoff replied that all he knew was that in riding right round the triangle, from A to B, from there to C and home to A,
his cyclometer registered exactly sixty miles, while Sliponsky could only say that he happened to know that C was exactly twelve miles from the road A to B — that is, to the point D,
as shown by the orange line.
Whereupon the General made a very simple calculation in his head and declared that the distance from A to B must be ______. Can the reader discover so easily how far it was?
Solution to the Problem:
The distance from A to B is 25 miles.
I don't see how the general was able to determine the distances in his head.
I needed to use several theorems from geometry to solve the problem.
Let a, b, c, d, and h be the lengths of the segments in the triangle where h = 12,
a + b + c + d = 60, and you are trying to find the length (c + d).
See the diagram below:
We know that a + b + c + d = 60 and that h = 12.
We know that a
2 + b
2 = (c + d)
2 (Pythagorean theorem).
We know that the altitude drawn to the hypotenuse of a right triangle divides it into two right triangles, each similar to the larger one.
Therefore, h is the geometric mean of c and d. So, h
2 = (c)(d) or in other words, c d = 144.
From the similar triangles, we can show that a
2= d (c + d) and b
2 = c (c + d).
From the perimeter formula above, we know that c + d = 60 - a - b.
We can rewrite it as c + d = 60 - (a + b)
Then squaring, we obtain (c + d)
2 = 3600 - 2(60)(a + b) + (a + b)
2
After putting everything together and solving, we obtain (c + d) = 25 miles.
a = 20, b = 15, c = 9, and d = 16.
The diagram below gives all the correct distances.
Everything checks. The three right triangles are 9 - 12 - 15, and 12 - 16 - 20, and 15 - 20 - 25.
The perimeter is
(a + b + c + d) = 20 + 15 + 9 + 16 = 60 miles.
