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:
Now there are lots of ways we could solve these two equations. Here's
- 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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