State and prove perpendicular axis theorem

Explanation

Perpendicular axis theorem : The moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of the moment of inertia about two perpendicular axis concurrent with perpendicular axis and lying in the plane of the body
`therefore I_z = I_x + I_y`
Proof: Consider a rectangular plane lamina. X and Y are two mutually perpendicular axis in the plane. Choose another perpendicular axis Z passing through the point 'O '.

Consider a particle Pin XOY plane.
Its co-ordinates are (x, y). Moment of inertia of particle about X-axis is `l_X =Sigma my62`
M.O.I about Y-axis is `l_y = Sigmamx^2`
M.O.I about Z axis is `I_Z = Sigma m. OP^2`
From triangle
`OP^2 = OA^2 + AP^2 =y^2 + x^2`
`therefore I_Z =Sigma m OP^2 = Sigma m (y^2 + x^2)`
`I_Z = Sigma my^2 + Sigma mx^2`
But `Sigma my^2 = I_x " and " Sigma mx^2 = I_y`
Moment of Inertia about a perpendicular axis passing through 'O' is ` I_Z = I_X + I_Y`
Hence perpendicular axis theorem is proved.