June 2018 Paper 1 Q7
7 Points \(A\), \(B\) and \(C\) lie on a circle, centre \(O\).
Angle \(AOC = x + 75^\circ\)
Angle \(ABC = 2x\)
Work out the value of \(x\). [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| reflex angle \(AOC = 2 \times 2x\) or \(4x\) | M1 | |
| their \(4x + x + 75 = 360\) | M1dep | oe If they start with this equation, the first M1, for reflex angle \(AOC = 4x\), is implied |
| \((x =)\ 57\) | A1 | |
| Alternative method 2 | ||
| reflex angle \(AOC = 360 - (x + 75)\) or \(285 - x\) | M1 | oe |
| \(360 - (x + 75) = 2(2x)\) or their \(285 - x = 2(2x)\) | M1dep | oe |
| \((x =)\ 57\) | A1 | |
| Alternative method 3 | ||
| angle at circumference \(= 180 - 2x\) | M1 | creating a cyclic quadrilateral |
| \(x + 75 = 2(180 - 2x)\) or \(x + 75 = 360 - 2(2x)\) | M1dep | oe |
| \((x =)\ 57\) | A1 | |
| Alternative method 4 | ||
| angle at circumference \(= \dfrac{x + 75}{2}\) | M1 | oe creating a cyclic quadrilateral |
| \(\dfrac{x + 75}{2} + 2x = 180\) | M1dep | oe \(\dfrac{x}{2} + \dfrac{\text{their } 75}{2} + 2x = 180\) scores this mark |
| \((x =)\ 57\) | A1 | |
Additional guidance
| \(4x = x + 75\) (ans \(x = 25\)) and \(x + 75 + 2x = 180\) (ans \(x = 35\)) both score 0 marks |