Circle Theorems: Eduqas GCSE Higher June 2023 Paper 1 Q18

Higher June 2023 Paper 1 Q18

EduqasHigherCurrent spec6 marksCircle Theorems

18.

(a)

Circle with centre O; points P, R and S on the circumference; radii OP and OS with reflex angle POS = 250°; chords PR and RS

Diagram not drawn to scale

The diagram shows a circle, centre O.
Points P, R and S are on the circumference of the circle.
Reflex angle \(P\hat{O}S = 250^\circ\).
\(O\hat{P}R : O\hat{S}R = 2 : 3\).

Calculate the size of \(O\hat{P}R\).
You must show all your working. [3]

(b)

Circle through A, B, C and D with AB parallel to DC; BC and AD extended meet at E; angles a° at A, b° at B, c° at C and d° at D inside ABCD; equal-length marks on BC and AD

Diagram not drawn to scale

ABE is an isosceles triangle.
AB is parallel to DC so that ABCD is an isosceles trapezium.

Prove that ABCD is always a cyclic quadrilateral.
State any reasons that you use in your proof. [3]

Circle with centre O, points P, R, S and a fourth point T on the circumference; dashed lines TP and TS; reflex angle 250° at O — Mark scheme (a)

Scheme	Marks	Comments
\(P\hat{R}S = 125(^\circ)\)	B1	Allow if marked on diagram
\(O\hat{P}R = \dfrac{2}{5}\times\big(360 - (125 + 110)\big)\) or \(125 - \dfrac{3}{5}\times125\) oe	B1
50(°)	B1	Accept 50 : 75. If no marks, award SC1 for correct evaluation of 2/5 × (250 – ‘their PRS’) provided 90 < PRS < 180.

Scheme	Marks	Comments
\(P\hat{T}S = 55(^\circ)\) AND \(P\hat{R}S = 125(^\circ)\)	B1	Allow if angles marked on diagram
\(O\hat{P}R = \dfrac{2}{5}\times\big(360 - (125 + 110)\big)\) or \(125 - \dfrac{3}{5}\times125\) oe	B1
50(°)	B1	Accept 50 : 75.

Mark scheme (b)

Scheme	Marks	Comments
\(a = b\) AND \(c = d\) (isosceles trapezium) \(a + d = 180^\circ\) OR \(b + c = 180^\circ\) (co-interior angles between parallel lines) \(a + c = 180^\circ\) OR \(b + d = 180^\circ\) Opposite angles in a cyclic quadrilateral (add up to 180°)	B3	Complete proof. At least two reasons must be provided. One of these reasons must be the cyclic quadrilateral. B2 for a complete proof with no reasons. B1 for \(a + c = 180^\circ\) OR \(b + d = 180^\circ\) AND \(a = b\) AND \(c = d\) OR \(a + d = 180^\circ\) OR \(b + c = 180^\circ\)
	(6)

Alternative method