Finding the dimensions of a rotated rectangle inside another rectangle.

[galneweinhaw]

Homework Statement

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?

[URL]http://bp3.blogger.com/_4Z2DKqKRYUc/Rnz_BgODzFI/AAAAAAAAAIw/uj_cVfPI8D4/s1600-h/Img_6-23-07_Blog.jpg
http://bp3.blogger.com/_4Z2DKqKRYUc/Rnz_BgODzFI/AAAAAAAAAIw/uj_cVfPI8D4/s1600-h/Img_6-23-07_Blog.jpg

If the image above is my example, I know the dimensions of ABCD and I know all the angles, such as < BPQ.

Homework Equations

Trig/Pythagorous...

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 =)

 

[diazona]

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...

 

[galneweinhaw]

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

 

[Mark44]

So show us what you've tried.

 

[galneweinhaw]

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.

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 + homework (areas)

 

[galneweinhaw]

eliminating x1,x2,y1,y2 I get...

XY = \frac{(w^2+h^2)tan\theta}{1+tan^2\theta} + hw

X = \frac{htan\theta + w}{\sqrt{1+tan^2\theta}}

Y = \frac{wtan\theta + h}{\sqrt{1+tan^2\theta}}

edit:
sub some trig identities

XY = (w^2+h^2)sin\theta cos\theta + hw

X = (htan\theta + w)cos\theta

Y = (wtan\theta + h)cos\theta

 

[galneweinhaw]

Further simplifying...

X = hsin\theta + wcos\theta

Y = wsin\theta + hcos\theta

LOL... I could have pulled that directly off the diagram! well, at tleast I know my algebra is sound =P

 

[igtech]

But with this, you find X and Y that it is supposed you already knew, what about finding h and w , huh??

 

[Tommaso_Russo]

Hi,

I've posted general solutions for all these kinds of problems in a new thread:
https://www.physicsforums.com/showthread.php?t=508715

sorry for the delay, it was not easy.

--
TRu-TS
Buon vento e cieli sereni