Buildings and Ladders

Submitted by Christopher Schoettle, 10 February 1997. Original article by Valerio De Angelis, this article by Allen Stenger.

We sent our Field Team out to place two ladders as shown in the figure, and to measure how high they reached on each building ( y and z in the diagram). The ladders are 15 meters and 20 meters long ( r = 20 and s = 15 ). The Field Team came back and reported that they had placed the ladders with no trouble, but they couldn't measure the heights. The longest tape measure they have is 6 meters, and they reported that the ladders definitely reached more than 6 meters, but they couldn't measure how high. They did measure very carefully the height where the ladders crossed ( h in the diagram), to a fraction of a centimeter, and they say that h is exactly 576 cm. They explained, "There are lots of triangles in this problem, we're sure there must be some way to figure out y and z from h .... No, we don't know how to do it, we forgot everything right after the exam, but there must be some clever person here who can figure it."

Are you the clever person they are thinking of? Can you figure y and z from the information given? If not, we can send the Field Team back out to measure some more stuff, as long as it can be done with a 6 meter tape; is there any additional measurement that would be helpful?

Hint 1

Yes, there really is enough information, but since you are reading this hint we'll assume you don't know how to do it yet. Let's imagine we send the Field Team out to make more measurements. The horizontal distance between buildings is definitely more than 6 m too, but they can measure that because it's on the ground (they can measure 6 m, mark the point, then move the tape and and continue measuring). If we know the distance x , how could we find y and z ?

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