This article covers the essential skills of circle properties – a core topic in DSE Paper 1 Section A(2) and Section B worth 4–7 marks. You will learn the basic definitions of circle elements (centre, radius, chord, arc, sector), the relationships between central angles, arcs, and chords, the inscribed angle theorem, the properties of cyclic quadrilaterals, and how to apply these theorems in geometric proofs. The article includes step-by-step worked examples, DSE exam techniques, practice questions. These are essential skills that appear regularly in DSE papers.
By the end of this article, you should be able to:
Circle geometry appears every year in DSE Paper 1 Section A(2) and Section B. Questions may ask you to:
These are essential skills that also appear in more advanced geometry topics.
Circle property questions often appear in Section A(2) as short-answer questions worth 3–4 marks, and in Section B as longer proof questions worth 5–7 marks.
A circle is a set of points equidistant from a fixed point called the centre. Key elements include:
In the same circle or equal circles:
Question: In a circle, central angle \( \angle AOB = 60^\circ \). Find the measure of arc \( AB \).
The measure of an arc is equal to its central angle.
Answer: Arc \( AB = 60^\circ \).
The inscribed angle theorem states:
Question: In the figure, \( \angle ABC = 40^\circ \). What is the measure of arc \( AC \)?
By the inscribed angle theorem, \( \angle ABC = \frac{1}{2} \times \text{arc } AC \).
So \( 40^\circ = \frac{1}{2} \times \text{arc } AC \) → \( \text{arc } AC = 80^\circ \).
Answer: Arc \( AC = 80^\circ \).
Theorem: An angle inscribed in a semicircle is a right angle (90°). That is, if \( AB \) is a diameter and \( C \) is on the circle, then \( \angle ACB = 90^\circ \).
Question: \( AB \) is a diameter of a circle. Point \( C \) is on the circle. If \( \angle CAB = 35^\circ \), find \( \angle ABC \).
Since \( AB \) is a diameter, \( \angle ACB = 90^\circ \).
In triangle \( ABC \), \( 35^\circ + 90^\circ + \angle ABC = 180^\circ \).
\( \angle ABC = 180^\circ - 125^\circ = 55^\circ \).
Answer: \( \angle ABC = 55^\circ \).
A cyclic quadrilateral is a quadrilateral whose vertices all lie on a circle. Key properties:
Question: In cyclic quadrilateral \( ABCD \), \( \angle A = 70^\circ \) and \( \angle C = 110^\circ \). Is this possible? Explain.
In a cyclic quadrilateral, opposite angles are supplementary.
\( \angle A + \angle C = 70^\circ + 110^\circ = 180^\circ \) ✓
Answer: Yes, this is possible since opposite angles sum to 180°.
Question: In the figure, \( O \) is the centre of the circle. \( \angle AOB = 100^\circ \). Find \( \angle ACB \).
\( \angle ACB \) is an inscribed angle intercepting arc \( AB \).
The central angle \( \angle AOB = 100^\circ \) intercepts the same arc.
By the inscribed angle theorem: \( \angle ACB = \frac{1}{2} \times 100^\circ = 50^\circ \).
Answer: \( 50^\circ \).
Question: In cyclic quadrilateral \( ABCD \), \( \angle A = 3x \) and \( \angle C = 2x + 20^\circ \). Find \( x \).
Opposite angles in a cyclic quadrilateral are supplementary.
\( 3x + (2x + 20^\circ) = 180^\circ \)
\( 5x + 20^\circ = 180^\circ \)
\( 5x = 160^\circ \) → \( x = 32^\circ \).
Answer: \( x = 32^\circ \).
Question: Prove that if \( ABCD \) is a cyclic quadrilateral, then \( \angle ABD = \angle ACD \).
In a cyclic quadrilateral, all four vertices lie on the same circle.
\( \angle ABD \) and \( \angle ACD \) are both inscribed angles that intercept the same arc \( AD \).
Inscribed angles intercepting the same arc are equal.
Answer: \( \angle ABD = \angle ACD \) (proved).
Question 1 MC
In a circle, central angle \( \angle AOB = 80^\circ \). What is the measure of arc \( AB \)?
A. \( 40^\circ \) B. \( 80^\circ \) C. \( 160^\circ \) D. \( 280^\circ \)
Question 2 MC
In the figure, \( O \) is the centre. \( \angle ABC = 35^\circ \). Find \( \angle AOC \).
A. \( 35^\circ \) B. \( 70^\circ \) C. \( 17.5^\circ \) D. \( 105^\circ \)
Question 3 Short Answer
In cyclic quadrilateral \( ABCD \), \( \angle A = 85^\circ \). Find \( \angle C \).
Question 4 Short Answer
\( AB \) is a diameter. \( C \) is on the circle. If \( \angle BAC = 28^\circ \), find \( \angle ABC \).
Question 5 MC
Which of the following is not a property of a cyclic quadrilateral?
A. Opposite angles are supplementary
B. Exterior angle equals opposite interior angle
C. All angles are equal
D. Vertices lie on a circle
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An Exercise on Circle Properties (Part 1) | Angles,Arcs & Chords