Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved a distance equal to half the longer side. Then the ratio of the shorter side to the longer side is:

ABCD is a rhombus with the diagonals AC and BD intersecting at the origin on the x - y plane. The equation of the straight line AD is x + y = 1. What is the equation of BC?

Consider a circle with unit radius. There are 7 adjacent sectors, S₁, S₂, S₃, ........, S₇ in the circle such that their total area is (1/8)^th of the area of the circle. Further, the area of the j^th sector is twice that of the (j-1)^th sector, for j = 2, .........., 7. What is the angle, in radians, subtended by the arc of S₁ at the centre of the circle?