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.
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Circle Properties (Part 2) | Tangents & Four Centres
Question 1 MC
In the figure, \( PA \) and \( PB \) are tangents to the circle. If \( \angle AOB = 120^\circ \), find \( \angle APB \).
A. \( 60^\circ \) B. \( 90^\circ \) C. \( 120^\circ \) D. \( 30^\circ \)
Question 2 MC
Which centre divides the medians in the ratio \( 2:1 \)?
A. Circumcentre B. Incentre C. Centroid D. Orthocentre
Question 3 Short Answer
A tangent to a circle at \( A \) makes an angle of \( 35^\circ \) with chord \( AB \). Find the angle in the alternate segment.
Question 4 Short Answer
Triangle \( ABC \) has vertices \( (0,0) \), \( (6,0) \), \( (0,8) \). Find the circumcentre.
Question 5 MC
The incentre of a triangle is always:
A. Outside the triangle B. Inside the triangle C. On the hypotenuse D. At the centroid
Question 6 Short Answer
In triangle \( ABC \), \( \angle A = 50^\circ \), \( \angle B = 60^\circ \). Find \( \angle BIC \) where \( I \) is the incentre.
Question 7 MC
Which centre is equidistant from the vertices of a triangle?
A. Circumcentre B. Incentre C. Centroid D. Orthocentre
Question 8 Short Answer
From an external point \( P \), two tangents \( PA \) and \( PB \) are drawn to a circle. If \( PA = 15 \) cm, find \( PB \).
Question 9 Short Answer
The incircle of triangle \( ABC \) touches \( AB \) at \( D \), \( BC \) at \( E \), \( CA \) at \( F \). If \( AD = 5 \), \( BE = 7 \), \( CF = 3 \), find the perimeter of \( ABC \).
Question 10 Short Answer
Prove that the tangents drawn from an external point to a circle are equal in length.
When using the alternate segment theorem, ensure you correctly identify the "alternate segment" – the angle in the opposite arc. Also, remember that the angle between the tangent and the chord can be on either side of the chord.