Higher June 2019 Paper 1 Q20

AQACurrent spec5 marksCircle Theorems

20 \(A\), \(B\) and \(C\) are points on a circle.

\(CD\) is a tangent.

Circle with points A, B and C on it, forming triangle ABC. D is outside the circle, joined to A and to C, with CD a tangent at C

Not drawn accurately

(a) Assume that triangle \(ABC\) is isosceles with \(\quad AC = BC\)

Prove that \(AB\) is parallel to \(DC\). [4 marks]

(b) In fact, triangle \(ABC\) is equilateral.

Tick the two boxes for the statements that must be correct. [1 mark]

\(AB\) is parallel to \(DC\)
\(AC\) bisects angle \(BCD\)
\(AC\) bisects angle \(BAD\)

Mark scheme (a)

Answer	Mark	Comments
Alternative method 1: shows that \(BAC = ACD\) and alternate angles
\(ACD = ABC\)	M1	accept both with same letter on diagram
\(ABC = BAC\)	M1	accept both with same letter on diagram
\(BAC = ACD\) and alternate segment (theorem) with M2 awarded	M1dep	dep on M2
Other two correct reasons given with M3 awarded	A1	eg (base angles of) isosceles triangle and alternate angles
Alternative method 2: shows that \(ABC + BCD = 180\) and co-interior angles
\(ACD = ABC\)	M1	accept both with same letter on diagram
\(ABC = BAC\)	M1	accept both with same letter on diagram
\(BCD = 180 - (BAC + ABC) + ACD\) and \(ABC + BCD = 180\) and alternate segment (theorem) with M2 awarded	M1dep	oe dep on M2
Other two correct reasons given with M3 awarded	A1	eg (base angles of) isosceles triangle and (co-)interior angles or allied angles
Alternative method 3: line from midpoint of \(AB\) to \(C\) is perpendicular to \(AB\) and \(CD\)
Let \(M\) be the midpoint of \(AB\) and \(MC\) is perpendicular to \(AB\)	M1	any letter
\(MC\) is perpendicular to \(CD\)	M1
\(AB\) and \(CD\) are both perpendicular to \(MC\) with M2 awarded	M1dep	oe dep on M2
Three correct reasons given with M3 awarded	A1	eg (perpendicular bisector of) isosceles triangle and \(MC\) goes through the centre of the circle and tangent is perpendicular to radius

Additional guidance

Other correct methods can be found by extending one or more of the lines. For example, by extending \(BC\) it is possible to use corresponding angles as a proof instead of alternating angles. This should be reflected in the reasons required for the last mark

In the scheme, \(ACD\) (for example) means angle \(ACD\) and not triangle \(ACD\)

Accept equality of angles indicated by labelling with the same letter, but not by arcs

Accept (angle) \(B\) for angle \(ABC\)
Do not accept (angle) \(A\) for angle \(BAC\) or (angle) \(C\) for angle \(ACB\) unless intention is clear from annotation of the diagram

For the third mark in alternative method 2, accept algebraic expressions for angles if clearly marked on the diagram

Do not award marks for an argument based only on assumed values of angles, but ignore 60° marked on diagram, which is for (b)

Ignore an angle marked at \(ADC\)

Ignore incorrect statements that do not affect the proof
eg \(ACD\) is an isosceles triangle (but not used in proof)