Math Problem: Dividing a square

[gentry_joey]

Here's a geometry puzzle to keep you occupied. The square pictured on the attachment has been divided into three pieces by making two parallel cuts. The three resulting pieces have equal areas. The middle piece (the almost-diagonal strip) is 10 inches wide. What is the area of the whole square?

Be sure to include your reasoning with your answer. Good luck!

[magwel]

I just have to say that your weekly math problems are really refreshing. It keeps us occupied when the work day is slow.

[gerald_weigel]

I think you both should be flogged repeatedly for enjoying his :P

[gentry_joey]

yah there kinda fun

[DVNT1]

Since no one seemed to appreciate my attempts of giving the answer plus hints last time, I attached the answer with notes written out.

See attached

Btw, AFAIK, it has to be assumed that the two cuts went to the midpoint on the relating side.

[nodnerb2]

Hi

Good One DVNT1.

This stuff used to make me cringe on exams. I just used to hope I had remembered enough other stuff to get by. Fortunately this was a process that worked in my case.

Cheers

Nodnerb2

[gentry_joey]

The area of the square is 1300 square inches.

Like most geometry problems, there are a lot of ways to do this one. Here is one short and elegant solution which makes use of the formula for the area of a parallelogram.

Label x, y, and z as in the figure. Now the area of a paralellogram is its height times its base width. The height of the yellow parallelogram is x and its base width is y. So, its area is xy. But this must also be one-third of the entire square, so


xy = (1/3)x2,
or


y = (1/3)x.
This means that the purple piece is a right triangle with legs of length x and x - y = (2/3)x and hypotenuse of length z. So, using the Pythagorean Theorem,


z2 = x2 + (4/9)x2 = (13/9)x2,
so


z = (1/3)sqrt(13)x.
Finally, we can look again at the yellow parallelogram, but this time turn our heads to the side just a little bit. Its "height" is now 10 and its "base width" is z = (1/3)sqrt(13)x, so its area (one-third the total area of the square) must be the product of these two numbers, so


(1/3)x2 = [10][(1/3)sqrt(13)x],
or


x = 10sqrt(13).
The area of the whole square is then


x2 = (10sqrt(13))2 = 1300 square inches.