Finding the dimensions of a rotated rectangle inside another rectangle.
1. The problem statement, all variables and given/known data
If I have a rectangle rotated at a known angle with respect to a rectangle of known dimensions that inscribes it, how can I find the dimensions of the inscribed/inner rectangle? http://bp3.blogger.com/_4Z2DKqKRYUc/Rnz_BgODzFI/AAAAAAAAAIw/uj_cVfPI8D4/s1600h/Img_62307_Blog.jpg http://bp3.blogger.com/_4Z2DKqKRYUc/...2307_Blog.jpg If the image above is my example, I know the dimensions of ABCD and I know all the angles, such as < BPQ. 2. Relevant equations Trig/Pythagorous... 3. The attempt at a solution I'll post if I come up with anything that looks like it's gettign anywhere =P Thanks for the help... let's see how my first ever post is received =) 
Re: Finding the dimensions of a rotated rectangle inside another rectangle.
Surely you must have tried something?
Hint: can you find the four triangles in the figure? From there, you have the trig formulas to calculate the lengths of the sides you need... 
Re: Finding the dimensions of a rotated rectangle inside another rectangle.
oh, I've been trying for a couple hours. But I haven't really made it anywhere =(

Re: Finding the dimensions of a rotated rectangle inside another rectangle.
So show us what you've tried.

Re: Finding the dimensions of a rotated rectangle inside another rectangle.
ok... I think I have something that should be able to go somewhere...
Here's a relabelled image: http://img7.imageshack.us/img7/764/rectb.jpg ɵ, X, and Y are known, trying to find h and w. y_{1}, y_{2}, x_{1}, x_{2}, w, and h are the unknowns (6) I can get seven equations: w^{2} = x_{2}^{2}+y_{1}^{2} h^{2} = x_{1}^{2}+y_{2}^{2} Y = y_{1} + y_{2} X = x_{1} + x_{2} y_{1} = x_{2 }tanɵ x_{1} = y_{2} tanɵ XY = x_{2}y_{1} + x_{1}y_{2} + hw (areas) 
Re: Finding the dimensions of a rotated rectangle inside another rectangle.
eliminating x1,x2,y1,y2 I get...
XY = [tex]\frac{(w^2+h^2)tan\theta}{1+tan^2\theta} + hw[/tex] X = [tex]\frac{htan\theta + w}{\sqrt{1+tan^2\theta}}[/tex] Y = [tex]\frac{wtan\theta + h}{\sqrt{1+tan^2\theta}}[/tex] edit: sub some trig identities XY = [tex](w^2+h^2)sin\theta cos\theta + hw[/tex] X = [tex](htan\theta + w)cos\theta[/tex] Y = [tex](wtan\theta + h)cos\theta[/tex] 
Re: Finding the dimensions of a rotated rectangle inside another rectangle.
Further simplifying...
[tex]X = hsin\theta + wcos\theta[/tex] [tex]Y = wsin\theta + hcos\theta[/tex] LOL.... I could have pulled that directly off the diagram! well, at tleast I know my algebra is sound =P 
Re: Finding the dimensions of a rotated rectangle inside another rectangle.
But with this, you find X and Y that it is supposed you already knew, what about finding h and w , huh??

Re: Finding the dimensions of a rotated rectangle inside another rectangle.
Hi,
I've posted general solutions for all these kinds of problems in a new thread: http://www.physicsforums.com/showthread.php?t=508715 sorry for the delay, it was not easy.  TRuTS Buon vento e cieli sereni 
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