Uniform plate of constant areal mass distribution

Diagram not to scale

Diagram not to scale

The figure above shows a uniform plate in the shape of a cyclic quadrilateral \(ABCO\) (with \(\angle B = \angle O = 90^{\circ}\) and side lengths \(AB\) and \(BC\) as mentioned). It is given that the sides \(OC\) and \(OA\) are equal.

The mass of the plate is \(7 \text{ kg}\) and its moment of inertia perpendicular to the plane of the plate and passing through its vertex \(O\) can be expressed as \(84 \lambda \text{ kg cm}^2\), where \(\lambda\) is a positive constant. Find the value of \(\lambda\).

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