November 2021 Paper 2 Q15
15 \(A\), \(B\) and \(C\) are points on a circle, centre \(O\).
\(ACD\) is a straight line.
Angle \(BCD = w\)
Not drawn accurately
Prove that \(\quad w = x + 90^\circ\) [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| angle \(ABO = x\) | M1 | may be seen on diagram implied by angle \(AOB = 180 - 2x\) |
| angle \(ACB = 180 - w\) | M1 | oe eg angle \(ACB + w = 180\) may be seen on diagram |
| angle \(AOB = 2 \times (180 - w)\) or angle \(AOB = 360 - 2w\) | M1dep | may be seen on diagram dep on 2nd M1 angle \(AOB\) may be seen as \(180 - 2x\) |
| \(x + x + 2 \times (180 - w) = 180\) | M1dep | oe eg \(2(180 - w) = 180 - 2x\) dep on M3 |
| \(w = x + 90\) with M4 and all reasons given | A1 | eg of reasons isosceles triangle and angles on a straight line and angle at centre and angle sum of triangle |
| Alternative method 2 | ||
| angle \(ABO = x\) | M1 | may be seen on diagram implied by angle \(AOB = 180 - 2x\) |
| angle \(AOB = 180 - x - x\) or angle \(AOB = 180 - 2x\) | M1dep | oe eg \(2x +\) angle \(AOB = 180\) may be seen on diagram |
| angle \(ACB = \dfrac{1}{2} \times (180 - x - x)\) or angle \(ACB = 90 - x\) | M1dep | oe eg angle \(ACB = \dfrac{1}{2} \times (180 - 2x)\) may be seen on diagram angle \(ACB\) may be seen as \(180 - w\) |
| \(\dfrac{1}{2} \times (180 - x - x) + w = 180\) | M1dep | oe eg \(w = 180 - (90 - x)\) |
| \(w = x + 90\) with M4 and all reasons given | A1 | eg of reasons isosceles triangle and angle sum of triangle and angle at centre and angles on a straight line |
| Alternative method 3 Draws tangent (eg \(PQ\)) at \(A\) | ||
| angle \(QAB = 90 - x\) | M1 | oe eg \(x +\) angle \(QAB = 90\) may be seen on diagram |
| angle \(ACB = 180 - w\) | M1 | oe eg angle \(ACB + w = 180\) may be seen on diagram |
| angle \(QAB =\) angle \(ACB\) | M1 | may be seen on diagram eg both angles labelled \(y\) |
| \(90 - x = 180 - w\) | M1dep | oe eg \(90 - x + w = 180\) dep on M3 |
| \(w = x + 90\) with M4 and all reasons given | A1 | eg of reasons radius perpendicular to tangent and angles on a straight line and alternate segment |
Additional guidance
| Allow angle \(BCD\) for \(w\) throughout | |
| 3rd M1 and 4th M1 may be seen in one line of working eg1 Alt 1 angle \(ABO = x\) angle \(ACB = 180 - w\) \(180 - 2x = 2 \times (180 - w)\) eg2 Alt 2 angle \(ABO = x\) angle \(AOB = 180 - 2x\) \(180 - w = \dfrac{1}{2} \times (180 - 2x)\) | M1 M1 M1M1 M1 M1 M1M1 |
| Condone slips in notation only if angles are marked in correct position on the diagram eg1 Do not allow angle \(C = 180 - w\) unless marked in correct position on the diagram eg2 Allow \(ACB\) for angle \(ACB\) | |
| For reasons, allow if the intention is clear eg1 Allow isos triangle for isosceles triangle eg2 Allow angles in a triangle for angle sum of a triangle eg3 Allow angles on a line for angles on a straight line | |
| For reasons do not allow incorrect statements eg do not allow angles in a triangle add to 360 |