After Tasha moves A to A′, A′ is 2 units to the right of A and 3 units above O. Therefore, A′=(sqrt2,sqrt3).
After Richard moves A′ to A′′, A′′ is 90∘ clockwise from A′ and the same distance from O. Therefore, A′′=(−sqrt3,sqrt2).
After Richard moves A to A1, A1 is 90∘ clockwise from A and the same distance from O. Therefore, A1=(−sqrt3,sqrt3).
After Tasha moves A1 to A2, A2 is 2 units to the right of A1 and 3 units above O. Therefore, A2=(−sqrt3+sqrst2,sqrt3+sqrt2).
Therefore, A′′A2=(−sqrt3+sqrt2−(−sqrt3))(−sqrt3+sqrt2−sqrt3+sqrt2) = sqrt(2).