Internet Math Challenge

Solution to the puzzle for 11 October 2002

The rectangle's sides measure squareroot(2) and 2*squareroot(2).

The key is to notice (see diagram at right) that the diagonals of both the square and the rectangle must be diameters of the circle. Since the area of the square is 5, we know that its side length is squareroot(5), and thus its diagonal (using the Pythagorean Theorem) is squareroot(10).

Now let x and y be the side lengths of the rectangle, as in the figure. Then we have two equations:

• x² + y² = 10 [the diagonal of the rectangle must have length squareroot(10)]
• xy = 4 [the area of the rectangle is 4]

• Adding the first equation to twice the second, we get x² + 2xy + y² = 10+8
  (x + y)² = 18
  x + y = squareroot(18) = 3*squareroot(2)
• But subtracting twice the second equation from the first, we get x² - 2xy + y² = 10-8
  (x - y)² = 2
  x - y = squareroot(2)
• Then, adding the two resulting equations we get (x + y) + (x - y) = 3*squareroot(2) + squareroot(2)
  2x = 4*squareroot(2)
  x = 2*squareroot(2) from which it follows easily that y = squareroot(2).

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