3. Square ABCD with vertices (0, 0), (2,0), (0, 2), and (2, 2) is rotated 30°around the origin. The area commonto the original and rotated squarescan be written in the form M/3. FindN.6/54/3O 3/2O 8/5O 5/3

Vertices of square ABCD: A(0,0); B(2,0); C(2,2), D(0,2) After rotation by 300 around origin, the new cordinate are Ax'Bx'Cx'Dx'Ay'By'Cy'Dy' = R02200022R = rotation matrixR = cosθ-sinθsinθcosθ, where θ = 30oR = 32-121232Ax'Bx'Cx'Dx'Ay'By'Cy'Dy' =32-12123202200022Ax'Bx'Cx'Dx'Ay'By'Cy'Dy' =033-1-1011+33Area of common region betwee two squares = square ABCD - area of green bounded region Area of region AB'EDA = area of ABCDA - area of ABFCEB'A (1) Now area of ABCDA = 2*2 = 4 units (2)area of ABFCEB'A = area of trapezium ABFB' ... + area of trapezium FCEB' (3) area of trapezium ABFB' = 12*BF(AB + B'F)area of trapezium ABFB' = 12*1*(2 + (2-3)) area of trapezium ABFB' = 12(4-3) (4)area of trapezium FCEB' = 12*FC(CE + B'F)area of trapezium FCEB' = 12*1*2-1.155 + (2-3) putting 1.155 = 23area of trapezium FCEB' = 12*1*2-23 + (2-3area of trapezium FCEB' = 124-53 (5)Putting (4) and (5) in (3) area of ABFCEB'A = 12(4-3) + 124-53 area of ABFCEB'A = 4 - 43 (6)Use (2) and (6) in (1)Area of region AB'EDA = 4 - 4 - 43 Area of region AB'EDA =43Area of region AB'EDA =43 3Area of region AB'EDA =43 3 = N3So N = 43