This article covers advanced circle properties including tangents and the four centres of a triangle – a high-frequency topic in DSE Paper 1 Section A(2) and Section B worth 4–7 marks. You will learn the tangent theorems (tangent perpendicular to radius, tangent lengths from an external point, and the alternate segment theorem), and the definitions and properties of the circumcentre, incentre, centroid, and orthocentre of a triangle, with applications in circle geometry. The article includes step-by-step worked examples, DSE exam techniques, practice questions. These are high-level skills that distinguish top-performing students.
By the end of this article, you should be able to:
Tangents and the four centres of a triangle appear every year in DSE Paper 1 Section A(2) and Section B. Questions may ask you to:
These are high-level skills that can help you secure top marks.
Tangent 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. The four centres are often tested in combination with circle theorems.
A tangent to a circle is perpendicular to the radius at the point of tangency.
If \( OT \) is a radius and \( PT \) is a tangent at \( T \), then \( OT \perp PT \).
From a point outside a circle, the two tangent lengths are equal.
If \( PA \) and \( PB \) are tangents from \( P \) to the circle, then \( PA = PB \).
The angle between a tangent and a chord is equal to the angle in the alternate segment (the angle subtended by the chord at the circumference on the opposite side).
If \( PT \) is a tangent at \( T \) and \( TA \) is a chord, then \( \angle PTA = \angle TBA \) where \( B \) is any point on the circle in the alternate segment.
Alternate segment theorem: The angle between the tangent and the chord equals the angle in the opposite arc (the "alternate" segment).
Question: In the figure, \( PT \) is tangent to the circle at \( T \), and \( OT = 5 \) cm, \( OP = 13 \) cm. Find the length of \( PT \).
By tangent-radius theorem, \( OT \perp PT \), so triangle \( OTP \) is right-angled.
Using Pythagoras: \( OP^2 = OT^2 + PT^2 \) → \( 13^2 = 5^2 + PT^2 \) → \( PT^2 = 169 - 25 = 144 \) → \( PT = 12 \) cm.
Answer: \( 12 \) cm.
Question: In the figure, \( PA \) is tangent at \( A \). \( \angle PAB = 40^\circ \). Find \( \angle ACB \).
By the alternate segment theorem, \( \angle PAB = \angle ACB \) (angle in the alternate segment).
Thus \( \angle ACB = 40^\circ \).
Answer: \( 40^\circ \).
A triangle has four important centres. Each has a distinct definition and geometric significance:
| Centre | Definition | Circle Associated |
|---|---|---|
| Circumcentre | Intersection of perpendicular bisectors of the sides | Circumcircle (passes through all vertices) |
| Incentre | Intersection of angle bisectors | Incircle (tangent to all sides) |
| Centroid | Intersection of medians | — |
| Orthocentre | Intersection of altitudes (perpendicular from vertex to opposite side) | — |
Don't confuse the circumcentre (equidistant from vertices) with the incentre (equidistant from sides). The circumcentre is the centre of the circumcircle, while the incentre is the centre of the incircle.
Question: Triangle \( ABC \) has vertices \( A(2,3) \), \( B(4,7) \), \( C(6,3) \). Find the circumcentre.
The circumcentre is the intersection of perpendicular bisectors. Since \( A \) and \( C \) have the same y-coordinate, the perpendicular bisector of \( AC \) is the vertical line \( x = \frac{2+6}{2}=4 \).
Midpoint of \( AB \) is \( (3,5) \), slope of \( AB \) is \( \frac{7-3}{4-2} = 2 \), so perpendicular slope = \( -\frac{1}{2} \). Equation: \( y-5 = -\frac{1}{2}(x-3) \). At \( x=4 \), \( y-5 = -\frac{1}{2}(1) = -0.5 \) → \( y=4.5 \). So circumcentre is \( (4, 4.5) \).
Answer: \( (4, 4.5) \).
Question: In triangle \( ABC \), \( \angle A = 60^\circ \), \( \angle B = 70^\circ \). Find \( \angle AIC \) where \( I \) is the incentre.
The incentre is the intersection of angle bisectors. In triangle \( AIC \), \( \angle IAC = \frac{1}{2}\angle A = 30^\circ \), \( \angle ICA = \frac{1}{2}\angle C \). First, \( \angle C = 180^\circ - 60^\circ - 70^\circ = 50^\circ \), so \( \angle ICA = 25^\circ \).
Then \( \angle AIC = 180^\circ - 30^\circ - 25^\circ = 125^\circ \).
Answer: \( 125^\circ \).
Question: Triangle \( ABC \) has vertices \( (1,2) \), \( (3,6) \), \( (5,4) \). Find the centroid.
Centroid is the average of the coordinates: \( \left( \frac{1+3+5}{3}, \frac{2+6+4}{3} \right) = \left( \frac{9}{3}, \frac{12}{3} \right) = (3,4) \).
Answer: \( (3,4) \).
In DSE questions, tangents and centres often appear together. For example, the incentre is the centre of the incircle, which is tangent to all sides. The circumcentre is the centre of the circumcircle, which passes through all vertices.
Question: In triangle \( ABC \), the incircle touches \( AB \) at \( D \). If \( AD = 4 \), \( DB = 6 \), and \( AC = 8 \), find the length of \( BC \).
Using tangent lengths from the same external point: from \( A \), tangents to the incircle are equal, so \( AD = AE = 4 \) (where \( E \) is touchpoint on \( AC \)). Since \( AC = 8 \), \( EC = 8 - 4 = 4 \).
From \( B \), tangents equal: \( BD = BF = 6 \) (where \( F \) is touchpoint on \( BC \)).
From \( C \), tangents equal: \( CE = CF = 4 \).
Thus \( BC = BF + FC = 6 + 4 = 10 \).
Answer: \( 10 \).
Question 1 MC
In the figure, \( PA \) and \( PB \) are tangents to the circle at \( A \) and \( B \). If \( \angle APB = 70^\circ \), find \( \angle AOB \) where \( O \) is the centre.
A. \( 70^\circ \) B. \( 110^\circ \) C. \( 140^\circ \) D. \( 55^\circ \)
Question 2 MC
Which centre of a triangle is equidistant from the three sides?
A. Circumcentre B. Incentre C. Centroid D. Orthocentre
Question 3 Short Answer
In the figure, \( PT \) is tangent at \( T \). \( \angle PTQ = 50^\circ \). Find \( \angle PRQ \).
Question 4 Short Answer
Triangle \( ABC \) has vertices \( (0,0) \), \( (4,0) \), \( (0,3) \). Find the circumcentre.
Question 5 MC
The incircle of triangle \( ABC \) touches \( BC \) at \( D \). If \( BD = 3 \), \( DC = 5 \), \( AB = 7 \), find \( AC \).
A. \( 9 \) B. \( 8 \) C. \( 7 \) D. \( 6 \)
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An Exercise on Circle Properties (Part 2) | Tangents & Four Centres