1D scaled rectangle bisecting a known rectangle

hi..

I need to fit a rectangle of fixed height (a) inside a known rectangle and have all four corners touch.

the top image is an example of a solved problem.. i don't need to know those numbers anymore-- i can figure them out.. but- i drew that one backwards (drew the white rectangle first then the black one around it) for sake of example..

so using the bottom image, how would i figure out the length of the white rectangle which fits inside the black one? (pretty obvious but the black rectangle is 5units x 8 and the rectangle needs to be 1 unit in one direction and can be scaled to length for the other dimension)

pretty much any relevant new info i can get for the white rectangle will be ok.. it doesn't have to be the length.. any of the angles will be ok too.

thanks for any help!

• no, it won't always be 1 unit.. it will however be a predetermined width.. that said.. 8.434 isn't the right number.. only two corners will touch instead of all 4..(well, i suppose i could get three corners to touch but still.. image at dropbox – j__e__f Jan 9 '15 at 7:52

So, noting that the 4 triangles were similar, you could solve a system of 3 unknowns and 3 equations. $$\frac{1}{AH}=\frac{L}{8-AE}\\\frac{1}{AE}=\frac{L}{5-AH}\\1^2=(AH)^2+(AE)^2$$