SOLUTION: There is a square-walled city with a door at the center of each of its sides. Find the dimensions of the city if there is a tree that is 30 yards due north of the northern gate th

Question 495900: There is a square-walled city with a door at the center of each of its sides. Find the dimensions of the city if there is a tree that is 30 yards due north of the northern gate that can just be seen beyond the corner of the city if one stands 750 yards due west of the western gate.
Ive drawn a picture of a square with a marker on each side halfway along the length of the side to denote the gates. Now i drew a line from the top of the square starting at the "gate" [perpendicular to the side] and labeled this line 30yards/tree. [due norther of northern gate] Then on the left side of the square i drew another line [again perpendicular to the side] and labeled this 750yards. [due west of western gate] Then i connected the ends of these two lines. However, i feel that i do not have enough information to then find the length of the sides. I tried denoting the length of the sides of the square city "x". [needing to find the length of the sides, i figured id solve for x somehow] this allowed me to see two small right triangles outside the "city" ; one with height 30 and base 1/2 x, the other with base 750 and height 1/2x. both of these triangles have an unknown hypotenuse and unknown x. Then i looked at the larger right triangle that would have height (1/2 x + 30) and base (1/2 x + 750) AGAIN, i have unknown hypotenuse and unknown x. I'm not sure if im looking at this completely wrong, or if im on the right tract somehow. I dont think the answer should be in terms of x. This is a problem from the Nine Chapters, the Chinese book. Gougu theorem [ie, pythagorean]

Any help or pointers or anything really, would be much appreciated.
Thanks in advance!
Answer by cleomenius(959) (Show Source): You can put this solution on YOUR website!
I do not believe you have enough information either.
What you do need to know is the length of the hypotenuse; or the distance from the two trees.
If you consider the distance from the wall to the center of the village, you could set the problem up as follows
( 30 + x)^2 + (750 + x)^2 = (distance between the trees)^2.
Here you have x, 2x = the length of the village.
So, (2x)^2 is the area of our city.
If there is a solution with the information given, I would like to here about it, if possible.
Cleomemius.
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