The answer
(a) congruent by RHS
(b) the two points where line \(TO\) produced meets the circle
O-Level E-Math 2019 Paper 1 Question 16 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 16 of the O-Level E-Math 2019 Paper 1. It tests tangent ⊥ radius, in the Circle properties / congruence area. It is worth 4 marks: 3 + 1. 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 \(OAT\) and \(OBT\): - \(\angle OAT = \angle OBT = 90^{\circ}\) (a tangent is perpendicular to the radius at the point of contact); - \(OA = OB\) (radii of the same circle); - \(OT = OT\) (common side, the hypotenuse of each right-angled triangle).
So \(\triangle OAT \equiv \triangle OBT\) by RHS (right angle, equal hypotenuses, equal sides).
(b) The figure is symmetrical about the line \(TO\). The two positions of \(P\) that make \(\triangle APT \equiv \triangle BPT\) are the points where the line through \(T\) and \(O\) cuts the circle, i.e. the point of the circle nearest \(T\) and the point furthest from \(T\) (both on line \(TO\)). Mark both.
Answer: (a) congruent by RHS
(b) the two points where line \(TO\) produced meets the circle
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 tangent ⊥ radius question from Circle properties / congruence, worth 4 marks: 3 + 1.
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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