Higher June 2023 Paper 2 Q20

AQACurrent spec3 marksCircle Theorems

20

(a) \(P\), \(Q\) and \(R\) are points on a circle.

\(S\) is a point inside triangle \(PQR\).

Circle with triangle PQR inscribed. S inside the triangle is joined to P and R; angle PSR = 130 degrees; angle PQR = x

Not drawn accurately

Assume that \(S\) is the centre of the circle.

Work out the size of angle \(x\). [1 mark]

(b) In fact, the centre of the circle is on \(PS\) but not at \(S\).

What does this mean about the size of angle \(x\)?

Tick one box. [1 mark]

  • It is the same as the answer to part (a)
  • It is greater than the answer to part (a)
  • It is smaller than the answer to part (a)
  • It is impossible to tell
(c) For a different circle,

\(AB\) is a tangent at \(A\)
\(C\) and \(D\) are on the circumference of the circle
\(AC = CD\)

Circle with points A, C and D on it and tangent AB at A. AC = CD (marked). Angle BAD = 70 degrees and angle CAD = y

Not drawn accurately

Here is Simon’s method to work out the size of angle \(y\).

Angle \(ADC = 70^\circ\)   (alternate segment theorem)
Therefore \(\quad y = 70^\circ\)   (angles in an isosceles triangle)

Is he correct?

Give a reason for your answer. [1 mark]

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