The answer
angle \(ABC = 90^{\circ}\) (angle in a semicircle)
O-Level E-Math 2023 Paper 1 Question 14 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 14 of the O-Level E-Math 2023 Paper 1. It tests angle in a semicircle, in the Circle properties / Angles & polygons area. It is worth 2 marks. 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.
Since \(D\) is equidistant from \(A\), \(B\), \(C\), we have \(DA = DB = DC\). So \(\triangle ABD\) is isosceles with \(\angle DAB = \angle DBA\), and \(\triangle BDC\) is isosceles with \(\angle DCB = \angle DBC\). The angles of \(\triangle ABC\) sum to \(180^{\circ}\): \[\angle DAB + \angle DCB + \angle ABC = \angle DBA + \angle DBC + \angle ABC = 2\angle ABC = 180°,\] so \(\angle ABC = 90^{\circ}\). *(Equivalently: \(D\) is the centre of the circle through \(A\), \(B\), \(C\) and, as \(A\), \(D\), \(C\) are collinear, \(AC\) is a diameter, so \(\angle ABC\) is the angle in a semicircle.)*
Answer: angle \(ABC = 90^{\circ}\) (angle in a semicircle)
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.
Want more questions like this, with worked solutions?
Join our mailing list and we will send practice sets and worked solutions. One email, no spam.
Same skill, different papers. Each has a verified worked solution.
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 angle in a semicircle question from Circle properties / Angles & polygons, worth 2 marks.
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.
Yes. Every worked solution here is free to read, with no sign-up wall.
Browse E-Math and A-Math by year in our worked-solutions library at /resources/solutions/o-level/.
Book a free trial and diagnostic. We will look at a real paper and show you exactly where the marks are going.