Two circles touch internally at P. O is the centre of the larger circle and OP is a diameter of the smaller circle. Prove that PA = AB.

7. Given: Two circles touch internally at P. O is the centre of the larger circle and OP is a diameter of the smaller circle.
Prove that PA = AB.

Step 1: Identify properties from the given information.

• O is the centre of the larger circle. P, A, B are points on the circumference of the larger circle. Therefore, OA, OB, and OP are all radii of the larger circle. $$OA = OB = OP$$
• OP is a diameter of the smaller circle. A is a point on the circumference of the smaller circle. The angle subtended by a diameter at any point on the circumference is $90^\circ$. Therefore, $\angle O