Circle Theorems: WJEC GCSE Higher November 2019 Unit 1 Q12

Higher November 2019 Unit 1 Q12

WJECHigherCurrent spec4 marksCircle Theorems

12. The points \(A\), \(B\), \(C\) and \(D\) lie on the circumference of a circle.
The straight line \(EDF\) is a tangent to the circle at \(D\).

\(B\widehat{A}C = x\) and \(C\widehat{D}E = 2x\), where \(x\) is measured in degrees.

Circle through B, A, D and C (in that order round the circle), with chords BA, AD, DC, CB and CA; tangent EDF touches the circle at D; angle BAC = x and angle CDE = 2x

Diagram not drawn to scale

Find the size of \(B\widehat{C}D\) in terms of \(x\).
Express your answer in its simplest form.
You must give a reason for each step of your solution. [4]

Mark scheme

Scheme	Marks	Comments
CAD = \(2x\)	B1	Check diagram. (If this is the only B mark awarded, then \(2x\) marked on diagram must be unambiguous. Otherwise, ignore spurious angles on diagram.)
(Reason =) Alternate segment (theorem)	E1	Dependent on B1. Allow ‘opposite segments’. Do not accept ‘alternate angles’ or ‘opposite angles’.
BCD = 180 – \(3x\) OR BCD = 3(60 – \(x\))	B1	F.T. 180 – (\(x\) + ‘their CAD’). Must be in simplest form. Mark final answer e.g. do not accept 60 – \(x\) or \(x\) = 60
(Reason =) Opposite angles in a cyclic quadrilateral (add up to 180°)	E1	If B0, E mark may be awarded provided there is a clear attempt to apply the circle theorem.