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Ok, so I have this homework problem and I want to know if I did it correctly and if there is an easier way to do so. Question goes:

There is a field in the shape of a square with an area of $1250$ meteres squared. In the center there is a sprinkler that sprays water. What is the shortest radius needed to completlely enclosed the square?

The way I did it was: I divided the square into four equal parts (each containing an area of $312.5$). Next I found the square root of the area of one individual square to get all sides of the square. Finally, I cut the square in half, forming a triangle, and did the Pythagorean theorem to get the hypotenuse (radius of the circle) and got it to be $25$. Did I do it correctly?

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    Your proof is correct. Here's another way of doing it. Find the side ($=\sqrt{1250} = 25\sqrt{2}$). Find the hypotenuse ($=25 \sqrt{2} \times \sqrt{2}=50$). Find half of the hypotenuse ($=25$). That gives the distance from the center of the square to any of the vertices, so that is the answer.2011-09-14

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Yes, this is correct. I'll try to explain this more clearly.

You're trying to determine the distance from the center of a square to a corner given its area. To begin, take the square root of the area of the square to determine the length of each side:

$ \sqrt{1250} = 25\sqrt{2} $

Then determine the length of the diagonal of the square using the Pythagorean theorem:

$ \sqrt{(25\sqrt{2})^2 + (25\sqrt{2})^2} = 50 $

One half of this length, 25 meters, is the distance between the center and a corner of the square. This is the radius of a circle that circumscribes the square.