Solve problems finding unknown angles and lengths and prove further results using the circle properties listed above.

A circle with centre $O$ has radius $8$ cm. Two chords $AB$ and $CD$ intersect at point $P$ inside the circle. The lengths are $AP = 6$ cm, $PB = 4$ cm, and $CP = 3$ cm. (a) Determine the length of $PD$. (1 mark) (b) Given that $\angle APD = 75^\circ$, determine the area of triangle $APD$ correct to two decimal places. (1 mark) (c) Hence determine the perpendicular distance from point $P$ to chord $CD$, correct to two decimal places. (1 mark)

Two circles have centres O and P respectively. The circles intersect at points A and B. The radius of circle O is 13 cm and the radius of circle P is 15 cm. The distance between the centres is OP = 14 cm. (a) Show that the length AB = 24 cm. (2 marks) (b) Hence, find the angle ∠AOB, correct to 1 decimal place. (1 mark) (c) Find the angle ∠APB, correct to 1 decimal place. (2 marks)