2026 NORTHBROOKS P1 Q9

2026 NORTHBROOKS P1 Q9

Secondary 4
6 marks

The diagram shows a circle, centre \(O\), passing through the vertices of triangle \(ABC\). The tangent \(TA\) meets the diameter \(AC\) at point \(A\). \(CB\) produced and \(AF\) produced meet at \(T\).

Prove that

  1. triangle \(BFA\) is similar to triangle \(ABC\),[2]
  2. \(AB^2=AC\times BF\),[1]
  3. \(AB^2+TB^2=(TC+AC)(TC-AC)\).[3]

Solution:

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Answer:(a) \(\triangle BFA\sim\triangle ABC\) by AA. (b) \(AB^2=AC\times BF\). (c) \(AB^2+TB^2=(TC+AC)(TC-AC)\).

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