   SEARCH HOME Math Central Quandaries & Queries  Question from Debbie, a parent: A park named Writer's Rectangle opened in town. When asked about the dimensions of the rectangle, the city planner, responded with these clues: ---The diagonals of the rectangular park plus its longer sides together measure seven times one of the shorter sides. ---The length of one diagonal is 250 m longer than one of the shorter sides. Use this information to find the area of the park. Hi Debbie.

Start with variables: Let S = the short side and L = the long side and D = the diagonal of the park.

Now turn the question into mathematical equations:

"The diagonals of the rectangular park plus its longer sides together measure seven times one of the shorter sides "

2D + 2L = 7S ( call this equation  )

"The length of one diagonal is 250 m longer than one of the shorter sides "

D = 250 + S ( equation  )

Finally, use what you know from Mr. Pythagoras:

D2 = L2 + S2 ( equation  )

Here you have three equations with three unknown variables, so you should be able to solve this set of simultaneous equations using the Elimination method or the Substitution method (or both if you wish).

I'd proceed as follows:

"hmm... only two equations have an L, so I'll eliminate the L by solving  for L and substituting that into "

 2L = 7S - 2D
 L = 7S/2 - D
 D2 = L2 + S2
 D2 = (7S/2 - D)2 + S2

So now we substitute the expression in  for D in :

 (250 + S)2 = (7S/2 - (250 + S))2 + S2

From here, you need to multiply it out, collect like terms and find the value(s) for S (throw out anything negative of course, because you won't have a negative length park!) using factoring or completing-the-square or the quadratic formula.

Then you can use that in  to find D and both of those to find L using . You can even check it using  at the end.

Hope this helps,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.