Thread: crossing supports

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)

What ideas have you had so far?

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.

I'd say you're not using similar triangles. How can you use similar triangles in this problem?

I can see I can setup the ratios: X/M = h2/L and X/(L-M) = h1/L

But these are what I have in my first 4 equations already... I'm not sure what else I should be looking for in regards to the similar triangles.

Those look right. Now try eliminating one of the variables you don't want, say L. What happens?