Chord and sector relationship

3 A sector, \(P O Q\), of a circle centre \(O\) has radius 7 cm and angle 1.7 radians (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{41d9ff74-82de-4ac5-928f-f6ab008319d2-2_469_723_662_712}

1. Find the length of the line \(P Q\).
2. Hence find the perimeter of the shaded area.

\includegraphics{figure_6} The diagram shows a metal plate made by removing a segment from a circle with centre \(O\) and radius \(8\) cm. The line \(AB\) is a chord of the circle and angle \(AOB = 2.4\) radians. Find

1. the length of \(AB\), [2]
2. the perimeter of the plate, [3]
3. the area of the plate. [3]

\includegraphics{figure_4} Fig. 4 shows sector OAB with sector angle 1.2 radians and arc length 4.2 cm. It also shows chord AB.

1. Find the radius of this sector. [2]
2. Calculate the perpendicular distance of the chord AB from O. [2]

\(P\) and \(Q\) are points on the circumference of a circle with centre \(O\) and radius \(r\). The angle \(POQ\) is \(\theta\) radians. Given that the chord \(PQ\) has length 4, find an expression for the length of the arc \(PQ\) in terms of \(\theta\) of only. [5]