1MA0 Higher June 2012 Paper 1 Q21
*21
Diagram NOT accurately drawn
\(B\), \(C\) and \(D\) are points on the circumference of a circle, centre \(O\).
\(AB\) and \(AD\) are tangents to the circle.
Angle \(DAB = 50^\circ\)
Work out the size of angle \(BCD\).
Give a reason for each stage in your working. (4)
| Working | Answer | Mark | Notes |
|---|---|---|---|
| \(ABO = ADO = 90^\circ\) (Angle between tangent and radius is 90°) \(DOB = 360 - 90 - 90 - 50\) (Angles in a quadrilateral add up to 360°) \(BCD = 130 \div 2\) (Angle at centre is twice angle at circumference) OR \(ABD = (180 - 50) \div 2\) (Base angles of an isosceles triangle) \(BCD = 65\) (Alternate segment theorem) | 65° | 4 | B1 for \(ABO = 90\) or \(ADO = 90\) (may be on diagram) B1 for \(BCD = 65\) (may be on diagram) C2 for \(BCD = 65^\circ\) stated or \(DCB = 65^\circ\) stated or angle \(C = 65^\circ\) stated with all reasons: angle between tangent and radius is 90°; angles in a quadrilateral sum to 360°; angle at centre is twice angle at circumference (accept angle at circumference is half (or \(\frac{1}{2}\)) the angle at the centre) (C1 for one correct and appropriate circle theorem reason) QWC: Working clearly laid out and reasons given using correct language OR B1 for \(ABD = 65\) or \(ADB = 65\) (may be on diagram) B1 for \(BCD = 65\) (may be on diagram) C2 for \(BCD = 65^\circ\) stated or \(DCB = 65^\circ\) stated or angle \(C = 65^\circ\) stated with all reasons: base angles of an isosceles triangle are equal; angles in a triangle sum to 180°; tangents from an external point are equal; alternate segment theorem (C1 for one correct and appropriate circle theorem reason) QWC: Working clearly laid out and reasons given using correct language |