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        riddles >> easy >> Geometry: Unique property of square
        
(Message started by: Aryabhatta on Dec 7^th, 2007, 1:07pm)

Title: Geometry: Unique property of square
Post by Aryabhatta on Dec 7^th, 2007, 1:07pm

Three copies of a rectangle R are placed sided by side and the angles a and b are defined as in the figure below.

If a + b = 45 degrees, show that R must be a square.

Title: Re: Geometry: Unique property of square
Post by Sir Col on Dec 7^th, 2007, 1:34pm

There's probably an easier method, but...

::
[hide]Let the rectangle measure x units wide by y units high.
So tan(a) = y/2x and tan(b) = y/3x.
tan(a+b) = (tan(a) + tan(b))/(1 – tan(a)tan(b)) = 5xy/(6x² – y²) = tan(45) = 1
6x² – 5xy – y² = 0
(6x + y)(x – y) = 0
Taking the positive solution, x = y.
Hence R is a square.
[/hide]::

A lovely problem, Aryabhatta

Title: Re: Geometry: Unique property of square
Post by Aryabhatta on Dec 7^th, 2007, 2:01pm

Thanks and well done Sir Col.

Also I forgot to mention... No using trigonometry!

(This problem was inspired by the first problem on this page: http://mathcircle.berkeley.edu/BAMA/BAMA99/BAMA99.html)

Title: Re: Geometry: Unique property of square
Post by ecoist on Dec 7^th, 2007, 8:43pm

Your link gave me a clue to avoiding trig, aryabhatta.

[hide]We may assume that the horizontal lengths of the rectangles are 1 and the vertical lengths are x>0. Form the grid with rectangles of horizontal sides 1 and vertical sides x. Let A be one of its points. When x=1, let B be the point 2 rectangles right and 1 rectangle up from A, and let C be the point 3 rectangles right and 1 rectangle down from A. Then AB and the horizontal line through A make an angle equal to a, and AC and the horizontal line through A make an angle equal to b. Hence angle BAC equals a+b. Further, AB and BC have equal length and are perpendicular. Hence angle BAC=45 degrees.

Now suppose x=/=1, and let B' and C' be the corresponding grid points. Then, when x>1, B' lies above B and C' lies below C, whence angle B'AC'>45 degrees. When x<1, then B' lies below B and C' lies above C, whence angle B'AC'< 45 degrees. Hence a+b=45 degrees only if x=1, and so the rectangles are squares.[/hide]

Title: Re: Geometry: Unique property of square
Post by Sir Col on Dec 8^th, 2007, 9:19am

That is a beautiful solution, ecoist; Euclid would be proud of you.

You made a small typo:
Further, AB and AC have equal length...

should read:
Further, AB and BC have equal length...

Title: Re: Geometry: Unique property of square
Post by ecoist on Dec 8^th, 2007, 9:40am

Thanks, Sir Col. I was wondering as I went to bed if I had made that typo! And what about Aryahbhatta, who put some extra meat on that cute little BLT in his link? Reminds me of a delicious burger called the "hangover"!

Title: Re: Geometry: Unique property of square
Post by Aryabhatta on Dec 8^th, 2007, 6:10pm

Well done ecoist. I had the exact same proof!

In fact if monotonicity does not convince people, then we can use your right triangle ABC and prove that ratio of sides of the rectangle must be 1.