4MA0 Higher June 2014 Paper 4R Q18
18
Diagram NOT accurately drawn
\(ABCD\) is the tangent at \(C\) to a circle, centre \(O\).
\(E\), \(F\) and \(G\) are points on the circle.
\(AEOG\) and \(BEF\) are straight lines.
Angle \(BAE = 36^\circ\)
Angle \(EBC = 70^\circ\)
(a)
(i) Find the size of angle \(AOC\).
(ii) Give reasons for your answer.
(2)(b) Find the size of angle \(CGF\). (3)
| Scheme | Marks |
|---|---|
| \(\angle AOC = 90 - 36\) Answer: 54 | B1 |
| Angle between tangent and radius is 90° | B1 |
| (2) |
Notes
B1: cao
B1: for Angle between tangent and radius is 90° (accept right-angle or perpendicular)
Using angle CGF = angle FGE + angle CGE
| Scheme | Marks |
|---|---|
| \(\angle FGE = 90 - 34\) (=56) | M1 |
\(\angle CGE = \text{“}54\text{”} \div 2\) (=27) or \(\dfrac{180 - (180 - \text{“}54\text{”})}{2}\) (=27) | M1 |
| 83 | A1 |
| (3) | |
| (5 marks) |
Notes
M1: may be on diagram
M1: may be on diagram
A1: cao
Alternative: using angles in a cyclic quadrilateral = 180°
Angle CGF = 180 − (angle CEO + angle FEG)
| Scheme | Marks |
|---|---|
| Angle CEO = (180 – “54”)÷2 (=63) and angle FEG = 180 – (180 – 70) – 36) (=34) | M1 |
| Angle CGF = 180 – (“34” + “63”) | M1 |
| 83 | A1 |
Notes
M1: may be on diagram
M1: dep on previous M1
A1: cao