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Right angles in a semicircle and a moving point

Right angles in a semicircle and a moving point

IP 2
7 marks

\(AB\) is a diameter of a circle with centre \(O\), and \(C\) is any other point on the circle.

  1. By joining \(OC\) and using isosceles triangles, prove that \(\angle ACB=90^\circ\).[3]
  2. Points \(A\) and \(B\) are fixed \(10\) cm apart. A point \(P\) moves so that \(\angle APB=90^\circ\). Describe the path of \(P\).[2]
  3. Find the greatest possible area of triangle \(APB\) and the position of \(P\) where it occurs.[2]

Solution:

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Answer:(a) Proven (b) The circle with diameter \(AB\) (centre its midpoint, radius \(5\) cm), excluding \(A\) and \(B\) (c) \(25\text{ cm}^2\), when \(PA=PB\)

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