Thursday, March 6, 2008
A Parallelogram Has Sides of Lengths 39 and 25 and a Diagonal of Length 34. So, What Makes It So Special!
Thanks to TC's inspired challenge to our readers in a comment on the Medians of a Triangle post, I've decided to expand it into an investigation for our readers and students (geometry with some trig needed).
Consider a parallelogram whose sides have lengths 39 and 25 and with one diagonal of length 34.
(a) Explain why this parallelogram is unique, i.e., all parallelograms with these characteristics are congruent. Why was it not necessary to specify that the 'shorter' diagonal was given?
(b) A parallelogram has sides of lengths a and b and diagonals of lengths c and d. Use the Law of Cosines to show that
c2 + d2 = 2(a2 + b2).
(c) Determine the length of the other diagonal. As an alternative, how would you do it without the formula in (b)?
(d) Determine the area of this parallelogram.
(e) So what makes this parallelogram unusual?
Comments:
(1) From the comments on the Medians post (Cotton Blossom and others), we know that we can construct rectangles and rhombuses whose sides and diagonals have integer lengths, but the above demonstrates a parallelogram that is neither of these special cases.
(2) The formula in (b) is not too difficult to prove, however, finding solutions to this Diophantine equation or a general solution is far more challenging!
(3) Note that the parallelogram in this challenge also has integral area. Finding other such parallelograms is not a simple exercise!
Posted by Dave Marain at 1:57 PM 10 comments
Labels: Diophantine equation, geometry, investigations, law of cosines