The answer
(a) angle \(ADC = (175 - 2x)^{\circ}\)
(b) \(x = 50\)
O-Level E-Math 2016 Paper 1 Question 10 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 10 of the O-Level E-Math 2016 Paper 1. It tests opposite angles of a cyclic quadrilateral, in the Circle properties area. It is worth 3 marks: 1 + 2. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a) \(ABCD\) is a cyclic quadrilateral, so opposite angles are supplementary: angle \(ADC + \) angle \(ABC = 180^{\circ}\). Hence angle \(ADC = 180^{\circ} - (2x + 5)^{\circ} = (175 - 2x)^{\circ}\).
(b) Angle \(AOC = 3x^{\circ}\) is the angle at the centre standing on arc \(AC\) (the arc on the \(B\) side). Angle \(ADC\) is the angle at the circumference standing on that same arc \(AC\), so by the angle-at-centre property angle \(ADC = \tfrac12 \times\) angle \(AOC = \tfrac12(3x)^{\circ} = (1.5x)^{\circ}\). Equating the two expressions for angle \(ADC\): \(1.5x = 175 - 2x \Rightarrow 3.5x = 175 \Rightarrow x = 50\). (Check: angle \(ABC = 2(50)+5 = 105^{\circ}\), angle \(ADC = 175 - 100 = 75^{\circ}\); \(105^{\circ} + 75^{\circ} = 180^{\circ}\) ✓, and \(\tfrac12(3 \times 50) = 75^{\circ}\) ✓.)
Answer: (a) angle \(ADC = (175 - 2x)^{\circ}\)
(b) \(x = 50\)
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a opposite angles of a cyclic quadrilateral question from Circle properties, worth 3 marks: 1 + 2.
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