The answer
proved (equal alternate angles \(\Rightarrow DE \parallel ST\))
O-Level A-Math 2020 Paper 2 Question 4 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 4 of the O-Level A-Math 2020 Paper 2. It tests tangent-chord (alternate segment) theorem, in the Proofs in plane geometry area. It is worth 5. 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.
Let \(\angle TAC = x\) (the angle between the tangent \(AT\) and the chord \(AC\); note \(E\) lies on \(AC\), so this is also \(\angle TAE\)).
- By the tangent-chord (alternate segment) theorem, \(\angle TAC = \angle ABC\) (the angle in the alternate segment). Since \(D\) lies on \(AB\), \(\angle ABC = \angle DBC\). Thus \(\angle DBC = x\). - \(BCED\) is a cyclic quadrilateral, so opposite angles are supplementary: \(\angle DBC + \angle DEC = 180^{\circ} \Rightarrow \angle DEC = 180^{\circ} - x\). - \(A\), \(E\), \(C\) are collinear, so \(\angle DEA = 180^{\circ} - \angle DEC = 180^{\circ} - (180^{\circ} - x) = x\).
Therefore \(\angle DEA = x = \angle TAE\). These are equal alternate angles for the lines \(DE\) and \(ST\) cut by the transversal \(AC\). Hence \(DE \parallel ST\). \(\blacksquare\)
Answer: proved (equal alternate angles \(\Rightarrow DE \parallel ST\))
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 A-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 tangent-chord (alternate segment) theorem question from Proofs in plane geometry, worth 5.
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