Finding the dimensions of a rotated rectangle inside another rectangle.

  1. 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/s1600-h/Img_6-23-07_Blog.jpg

    [​IMG]

    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 =)
     
  2. jcsd
  3. diazona

    diazona 2,156
    Homework Helper

    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...
     
  4. oh, I've been trying for a couple hours. But I haven't really made it anywhere =(
     
  5. Mark44

    Staff: Mentor

    So show us what you've tried.
     
  6. ok... I think I have something that should be able to go somewhere...

    Here's a relabelled image:
    [​IMG]

    ɵ, X, and Y are known, trying to find h and w.

    y1, y2, x1, x2, w, and h are the unknowns (6)

    I can get seven equations:

    w2 = x22+y12

    h2 = x12+y22

    Y = y1 + y2

    X = x1 + x2

    y1 = x2 tanɵ

    x1 = y2 tanɵ

    XY = x2y1 + x1y2 + hw (areas)
     
  7. 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]
     
    Last edited: May 2, 2009
  8. 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
     
  9. But with this, you find X and Y that it is supposed you already knew, what about finding h and w , huh??
     
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