The answer
(a) proven
(b) proven
O-Level A-Math 2021 Paper 1 Question 11 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 11 of the O-Level A-Math 2021 Paper 1. It tests alternate segment theorem, in the Proofs in plane geometry area. It is worth 7 marks: 4 + 3. 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) The tangent \(CT\) touches the circle at \(C\), and \(CA\) is a chord, so by the alternate segment theorem \(\angle ACT = \angle ABC\) (angle in the alternate segment). Because \(BA \parallel CT\) with \(AC\) a transversal, \(\angle BAC = \angle ACT\) (alternate angles). Combining, \(\angle BAC = \angle ACT = \angle ABC\). With two equal angles, triangle \(ABC\) is isosceles. (proven)
(b) Let \(\angle BAC = \angle ABC = x\). The angle sum of triangle \(ABC\) gives \(\angle BCA = 180^{\circ} - 2x\). In triangle \(ACT\), \(\angle ACT = x\) (shown above), and \(\angle CAT = \angle ACT = x\) because the tangents \(TA\) and \(TC\) from the external point \(T\) are equal (\(TA = TC\), so triangle \(TAC\) is isosceles). The angle sum of triangle \(ACT\) then gives \(\angle CTA = 180^{\circ} - 2x\). Therefore \(\angle BCA = \angle CTA\). (proven)
Answer: (a) proven
(b) proven
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 alternate segment theorem question from Proofs in plane geometry, worth 7 marks: 4 + 3.
Yes. IP (Integrated Programme) schools teach the same O-Level Mathematics content; they just sequence it differently and set their own internal exams, so these worked solutions apply to IP students too.
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