2026 AHS P1 Q15

2026 AHS P1 Q15

Secondary 4
7 marks

The diagram shows a quadrilateral, \(ABCD\), inscribed in a circle. The lines \(SAR\) and \(TCS\) are tangents to the circle at \(A\) and \(C\) respectively. The lines \(ADU\) and \(SCT\) are parallel.

  1. Prove that angle \(ACD=\) angle \(ASC\).[2]
  2. Prove that angle \(ABC=\) angle \(UDC\).[2]
  3. Given that \(A\), \(S\) and \(C\) lie on a circle, and \(B\) is the centre of this circle, prove that the triangle \(ASC\) is equilateral.[3]

Solution:

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Answer:(a) \(\angle ACD=\angle ASC\). (b) \(\angle ABC=\angle UDC\). (c) All three angles of \(\triangle ASC\) are \(60^\circ\), so the triangle is equilateral.

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