What ideas have you had so far?
Any and all help is appreciated. I've been working this problem all day and at this point I keep going around in circles...
Consider two buildings with heights h1 and h2, and an alleyway between them of length L. Suppose that two supports are attached in the manner shown below.
Yfrog Image : yfrog.com/0strigproblemp
We call the height at which the supports cross, X. We divide L into two parts: M and L-M. Using similar triangles with h1, h2, X, M, and L-M, show that the height of the crossing, X, is given by the equation: x = (h1*h2)/(h1+h2)
Well the first thing I do is assign a variable value to the angles in the lower left and lower right, y and z respectively.
I then write 4 equations from this:
tan(y) = (h2/L)
tan(y) = (X/M)
tan(z) = (h1/L)
tan(z) = [X/(L-M)]
Then I'll solve for different variables. I feel like I'm stumbling in the dark at this point but here's some of my work...
tan(y) = (X/M) tan(z) = [X/(L-M)]
X = Mtan(y) X = (L-M)tan(z)
Mtan(y) = (L-M)tan(z)
Mtan(y) = Ltan(z) - Mtan(z)
Mtan(y) + Mtan(z) = Ltan(z)
M[tan(y) + tan(z)] = Ltan(z)
M = Ltan(z) / [tan(y) + tan(z)]
Then I'll plug this value back into one of my 4 starting equations and say...
tan(y) = (X/M) = [X/ (Ltan(z)/[tan(y) + tan(z)])]
tan(y) = (h2/L)
(h2/L) = [X/ (Ltan(z)/[tan(y) + tan(z)])]
And again I fumble around trying to manipulate the equations, but I don't feel like I'm getting any closer or on the right track to the proof I need.