SOLUTION: A farmer sold a square shaped acre with an oak tree outside each corner. The buyer later returned to buy another acre. The farmer was eager to sell, but wanted to retain ownership
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Question 183969: A farmer sold a square shaped acre with an oak tree outside each corner. The buyer later returned to buy another acre. The farmer was eager to sell, but wanted to retain ownership of the oaks too. The tough part is that the buyer wanted the whole parcel to be square. In other words, you need to find a way to double the size of the plot and keep it square while at the same time leaving the farmer's four oak trees outside the parcel boundaries. Explain your answer in detail. Answer by solver91311(24713) (Show Source):
Draw a line that passes through one corner of the orginal acre plot at a 45 degree angle to an adjacent side of the original acre plot. Repeat this process at each corner. The four lines will intersect in four points forming the corners of a new, larger square rotated 45 degrees from the original.
If you assume that the sides of the original square were 1 unit in length, then the area of the original square is 1. Superimposing the new, larger square over the original square creates four isoceles right triangles with hypotenuse measuring 1. Each leg of these triangles is therefore (Proof left as an exercise for the student). Each leg of one of these triangles forms one-half of the side of the new square, so the length measure of the side is and therefore the area is 2, exactly twice the are of the original square.
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