The diagram shows a large semicircle with diameter    and two smaller semicircles with diameters    and  , respectively, where    is a point on the diameter  . The point    is the centre of the semicircle with diameter  .

The line perpendicular to    through    meets the largest semicircle at the point  . The points    and    are the intersections of the lines    and    with the smaller semicircles. The point    is the intersection of the lines    and  .
 

Explain why is a rectangle.   (3 marks)

In the diagram, the points , , and are on the circumference of a circle, whose centre lies on . The chord intersects the diameter at . The tangent at passes through the point .

It is given that    and  .

1. What is the size of  ?   (1 mark)
   

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2. What is the size of  ?   (2 marks)
   

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In the diagram, is a diameter of the circle centred and .
 

Find .  (2 marks)