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\includegraphics[max width=\textwidth, alt={}, center]{80a20f05-61db-42d9-b4ba-53eea2290b2d-10_780_814_264_662} The diagram shows a symmetrical metal plate. The plate is made by removing two identical pieces from a circular disc with centre \(C\). The boundary of the plate consists of two \(\operatorname { arcs } P S\) and \(Q R\) of the original circle and two semicircles with \(P Q\) and \(R S\) as diameters. The radius of the circle with centre \(C\) is 4 cm , and \(P Q = R S = 4 \mathrm {~cm}\) also.

1. Show that angle \(P C S = \frac { 2 } { 3 } \pi\) radians.
2. Find the exact perimeter of the plate.
3. Show that the area of the plate is \(\left( \frac { 20 } { 3 } \pi + 8 \sqrt { 3 } \right) \mathrm { cm } ^ { 2 }\).