Practice › Measures, Shape and Space › Centres of Triangles in the Coordinate Plane

Centres of Triangles in the Coordinate Plane: Paper 1 practice

Five long questions that find the four centres of a triangle with the tools of coordinate geometry: medians, perpendicular bisectors, altitudes and angle bisectors. The set opens gently and ends with three centres of one triangle on a single line. Answers sit under each question. Detailed solutions can be requested by email.

5 questions · about 35 minutes · answers below each question

All questions are original, written for Math and AI Academy and modelled on past-paper style. They are not reproductions of HKEAA examination questions.

Question 1 · Modelled on HKDSE 2013 Paper 2 Q42 — moved to Paper 1 and extended: the centroid feeds a perpendicular line instead of being the final answer

The coordinates of the points

A

B

and

C

are

(- 2, 5)

(4, - 3)

and

(10, 7)

respectively. Let

G

be the centroid of

△ A B C

(a) Find the coordinates of $G$ . (2 marks)

(b) Find the equation of the straight line which passes through $G$ and is perpendicular to $B C$ . (2 marks)

Show answer

(a) $G = (4, 3)$
(b) $3 x + 5 y - 27 = 0$

Question 2 · Modelled on HKDSE 2022 Paper 2 Q41 — the circumcentre is built step by step from perpendicular bisectors rather than read from a condition

The coordinates of the points

A

B

and

C

are

(2, 8)

(- 4, 2)

and

(8, - 4)

respectively. Let

K

be the circumcentre of

△ A B C

(a) Find the equation of the perpendicular bisector of $A B$ . (2 marks)

(b) Find the coordinates of $K$ . (2 marks)

(c) Find the distance from $K$ to $A$ . (1 mark)

Show answer

(a) $x + y - 4 = 0$
(b) $K = (3, 1)$
(c) $52$

Question 3 · Modelled on HKDSE 2015 Paper 2 Q42 — the orthocentre is found through two altitudes, then its position is judged

The coordinates of the points

P

Q

and

R

are

(0, 6)

(- 3, 0)

and

(9, 0)

respectively. Let

H

be the orthocentre of

△ P QR

(a) Find the equation of the altitude of $△ P QR$ passing through $Q$ . (2 marks)

(b) Find the coordinates of $H$ . (2 marks)

(c) Does $H$ lie inside $△ P QR$ ? Explain your answer. (1 mark)

Show answer

(a) $3 x - 2 y + 9 = 0$
(b) $H = (0, \frac{9}{2})$
(c) Yes. $H$ lies on the altitude from $P$ , between $P (0, 6)$ and its foot $(0, 0)$ on $QR$ .

Question 4 · Modelled on HKDSE 2025 Paper 2 Q41 — the in-centre of a right-angled triangle, found by the area method and then used to draw an angle bisector

Let

O

be the origin. The coordinates of the points

A

and

B

are

(24, 0)

and

(0, 10)

respectively. Let

I

be the in-centre of

△ O A B

(a) Find the equation of $A B$ . (1 mark)

(b) Explain why $I$ lies on the straight line $y = x$ . (1 mark)

(c) By considering the area of $△ O A B$ , find the coordinates of $I$ . (3 marks)

(d) Find the equation of the straight line passing through $B$ and $I$ . (1 mark)

Show answer

(a) $5 x + 12 y - 120 = 0$
(b) $I$ lies on the bisector of $∠ A O B = 9 0^{\circ}$ , which is the line $y = x$ .
(c) $I = (4, 4)$
(d) $3 x + 2 y - 20 = 0$

Question 5 · Modelled on HKDSE 2023 Paper 1 Q19 — extended with added structure: three centres of one triangle, then a collinearity and a ratio to prove

The coordinates of the points

A

B

and

C

are

(- 6, 0)

(12, 0)

and

(0, 12)

respectively. Let

H

K

and

G

be the orthocentre, the circumcentre and the centroid of

△ A B C

respectively.

(a) Find the coordinates of $H$ . (2 marks)

(b) Find the coordinates of $K$ . (2 marks)

(c) Write down the coordinates of $G$ . Hence, prove that $H$ , $G$ and $K$ are collinear, and find $H G : G K$ . (3 marks)

Show answer

(a) $H = (0, 6)$
(b) $K = (3, 3)$
(c) $G = (2, 4)$ ; slope of $H G$ = slope of $G K$ = $- 1$ , so the three points are collinear; $H G : G K = 2 : 1$

Continue practising this topic: Try the Paper 1 Challenge for harder variants, or move to Paper 2 multiple-choice — also available as Paper 2 Challenge MC.

Where this sits in the syllabus

Centres of triangles are first met in junior forms as constructions. In HKDSE Paper 1 they return in coordinates, usually in Section B, where each centre is found as the intersection of two straight lines. Work the Equations of Straight Lines sets first if slopes and perpendicular lines are not yet automatic.

Full coverage of this topic, and the rest of the course, is on the Mathematics page. More sets are listed on the practice index.

Start with a free discovery session

No cost, no obligation. We look at where your child actually is, and show you what a term of real progress would look like.

Book a free discovery session See the programmes