The reflex angle of AEC is 270 and so by the ‘angle at the centre’ angle theorem angle ABC must be half that, 135 degrees. Therefore angle ABD is 45 and BAD is also 45. Since ABD is an isosceles right triangle with hypotenuse root 2, AD = BD = 1. The right triangle ACD has legs equal to 1 and 7, and so by Pythagoras, AC is equal to the square root of 50. But AC is also the hypotenuse of the isosceles right triangle ACE, whose legs are the radius of the quarter circle we are seeking. Since 2r^2 = 50, r^2 = 25 and so the radius r is equal to 5.