The answer
(a) similar by AA
(b) a trapezium
O-Level A-Math 2023 Paper 2 Question 2 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 2 of the O-Level A-Math 2023 Paper 2. It tests tangent-chord (alternate segment), in the Proofs in plane geometry area. It is worth 9 marks: 4 + 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.
(a) In triangles \(ABD\) and \(AFB\): - \(\angle BAD = \angle FAB\) (the same angle at \(A\), since \(A\), \(F\), \(D\) are collinear on the secant); - \(\angle ADB = \angle ABF\) (tangent-chord: the angle between tangent \(AB\) and chord \(BF\) equals the angle \(\angle BDF\) in the alternate segment).
With two pairs of equal angles, triangle \(ABD\) is similar to triangle \(AFB\) (AA), with correspondence \(A \leftrightarrow A\), \(B \leftrightarrow F\), \(D \leftrightarrow B\).
(b) \(FD\) is a diameter, so \(\angle FBD = 90^{\circ}\) (angle in a semicircle), i.e. \(BF \perp BD\). The circle through \(B\), \(C\), \(X\) has \(BC\) as diameter, so \(\angle BXC = 90^{\circ}\) (angle in a semicircle); since \(X\) lies on \(BD\) and on \(EC\), this means \(EC \perp BD\). Therefore \(BF\) and \(EC\) are both perpendicular to \(BD\), so \(BF \parallel EC\). The quadrilateral \(EBFC\) has exactly one pair of parallel sides (\(BF \parallel EC\); \(EB\) and \(FC\) are not parallel), so it is a trapezium.
Answer: (a) similar by AA
(b) a trapezium
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) question from Proofs in plane geometry, worth 9 marks: 4 + 5.
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