Higher June 2022 Paper 3 Q18
18 \(A\), \(B\) and \(C\) are points on a circle, centre \(O\).
\(DC\) is a tangent to the circle.
Not drawn accurately
Show that angle \(ABO\) : angle \(ACO\) = 3 : 1 [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 – using angles around \(O\) and angles inside arrowhead | ||
| \(ACO\) = 90 − 83 or \(ACO\) = 7 | M1 | may be seen on diagram |
| Acute \(BOC\) = 2 × 28 or acute \(BOC\) = 56 | M1 | may be seen on diagram |
| Reflex \(BOC\) = 360 − their 56 or reflex \(BOC\) = 304 | M1dep | may be seen on diagram dep on 2nd M1 |
| \(ABO\) = 360 − their 304 − their 7 − 28 or \(ABO\) = 21 | M1dep | may be seen on diagram dep on M3 |
| \(ABO\) = 21 and \(ACO\) = 7 and 21 : 7 = 3 : 1 | A1 | all angle values must be seen |
| Alternative method 2 – with line \(OA\) added | ||
| \(ACO\) = 90 − 83 or \(ACO\) = 7 | M1 | may be seen on diagram |
| \(OAC\) = 7 or \(ABO\) + \(ACO\) = 28 | M1dep | may be seen on diagram |
| \(OAB\) = 28 − 7 or \(OAB\) = 21 or \(ABO\) = 28 − 7 | M1dep | may be seen on diagram dep on M2 |
| \(ABO\) = 21 | M1dep | may be seen on diagram dep on M3 |
| \(ABO\) = 21 and \(ACO\) = 7 and 21 : 7 = 3 : 1 | A1 | all angle values must be seen |
| Alternative method 3 – using alternate segment theorem | ||
| \(ACO\) = 90 − 83 or \(ACO\) = 7 | M1 | may be seen on diagram |
| Acute \(BOC\) = 2 × 28 or acute \(BOC\) = 56 | M1 | may be seen on diagram |
| \(ABC\) = 83 | M1 | may be seen on diagram |
| \(OBC\) = \(\dfrac{180 - \text{their } 56}{2}\) or \(OBC\) = 62 and \(ABO\) = 83 − their 62 or \(ABO\) = 21 | M1dep | may be seen on diagram, dep on 2nd and 3rd M1 |
| \(ABO\) = 21 and \(ACO\) = 7 and 21 : 7 = 3 : 1 | A1 | all angle values must be seen |
| Alternative method 4 – using triangles \(OBC\) and \(ABC\) | ||
| \(ACO\) = 90 − 83 or \(ACO\) = 7 | M1 | may be seen on diagram |
| Acute \(BOC\) = 2 × 28 or acute \(BOC\) = 56 | M1 | may be seen on diagram |
| \(OBC\) = \(\dfrac{180 - \text{their } 56}{2}\) or \(OBC\) = 62 | M1dep | may be seen on diagram or angle \(OCB\) dep on 2nd M1 |
| \(ABO\) = 180 − 28 − 62 − 62 − 7 or \(ABO\) = 21 | M1dep | oe may be seen on diagram dep on M3 |
| \(ABO\) = 21 and \(ACO\) = 7 and 21 : 7 = 3 : 1 | A1 | all angle values must be seen |
Additional guidance
| If angles are not correctly positioned on the diagram they must be correctly identified in the working, eg \(BOC\) = 56 is M0 if not correctly positioned on the diagram and not identified as acute | |
| \(ACO\) = 7 and \(ABO\) : \(ACO\) = 21 : 7 with no other correct working | M1M0M0M0A0 |