Question 525009: If the perimeter of a rectangle is 27 cm. Find the dimension for which the diagonal is as short as possible
Answer by lmeeks54(111)   (Show Source):
You can put this solution on YOUR website! You can solve this with logic or with numbers.
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First with logic: Comparing rectangles and squares, the longer the rectangle (i.e., the greater the difference between the length of the long side and the length of the short side), the longer the diagonal. Conversely, the "squarer" the rectangle, the shorter the diagonal. That is, as the rectangle converges to becoming a square (a specific case of a rectangle in which the long side = the short side), the diagonal gets shorter until the square is achieved.
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So, the easiest way to solve this is to consider the case of the square, where: Perimeter, P = 27 cm, and each side, S = 27/4 cm. S = 6.75 cm
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Therefore, the answer to the question is:
rectangle (square) of dimensions: 6.75 cm x 6.75 cm
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cheers,
Lee
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To continue on to compute the diagonal, "c", the Pythagorean Theorem applies:
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c^2 = a^2 + b^2 ... or c^2 = a^2 + a^2, or c^2 = 2a^2
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c = sqrt(2a^2)
c = sqrt(2*6.75^2)
c = ~9.546
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Interestingly, the rectangle (square) with the shortest diagonal also has the greatest area, A = ~45.563 cm^2
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