how did you know to use radical 2 on the triangle to find x and y 

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Pythagorean Theorem and given isosceles triangle with equal sides and angles of 45 degrees each:  x^2 + x^2 = r^2.  Assume length of side is 1 then r^2 = 1^2 + 1^2 or r^2 =2.  r is then equal to radical 2.