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Centres of Triangles in the Coordinate Plane: Paper 2 multiple-choice

Five multiple-choice questions in the style of Section B of HKDSE Paper 2, where a triangle centre in coordinates appears almost every year. Several distractors are the right answer for a different centre, so name the centre before you calculate. Answers sit under each question. Detailed solutions can be requested by email.

5 questions · about 8 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 — reversed: the centroid is given and a vertex is found

The coordinates of the points

A

and

B

are

(- 3, 5)

and

(7, - 1)

respectively. If the centroid of

△ A B C

(2, 3)

, then the coordinates of

C

are

$(- 2, - 5)$
$(4, 2)$
$(2, 5)$
$(6, 9)$

Show answer

C. $C = (3 \times 2 - (- 3) - 7, 3 \times 3 - 5 - (- 1)) = (2, 5)$

Question 2 · Modelled on HKDSE 2013 Paper 2 Q42 — the circumcentre of a triangle with no side on the y-axis

Let

O

be the origin. The coordinates of the points

A

and

B

are

(10, 0)

and

(4, 8)

respectively. The coordinates of the circumcentre of

△ O A B

are

$(5, \frac{5}{2})$
$(\frac{14}{3}, \frac{8}{3})$
$(4, 3)$
$(5, 4)$

Show answer

A. The circumcentre lies on $x = 5$ ; equal distances to $O$ and $B$ give $y = \frac{5}{2}$ .

Question 3 · Modelled on HKDSE 2015 Paper 2 Q42 — asks for both coordinates, with the other centres as distractors

Let

O

be the origin. The coordinates of the points

P

and

Q

are

(8, 0)

and

(2, 6)

respectively. The coordinates of the orthocentre of

△ O P Q

are

$(4, 2)$
$(\frac{10}{3}, 2)$
$(2, 4)$
$(2, 2)$

Show answer

D. The altitude from $Q$ is $x = 2$ ; the altitude from $O$ is $y = x$ .

Question 4 · Modelled on HKDSE 2025 Paper 2 Q41 — combines the in-centre and the circumcentre of one right-angled triangle

Let

O

be the origin. The coordinates of the points

A

and

B

are

(16, 0)

and

(0, 12)

respectively. Find the distance between the in-centre and the circumcentre of

△ O A B

$4$
$25$
$10$
$213$

Show answer

B. In-centre $(4, 4)$ , circumcentre $(8, 6)$ , the mid-point of $A B$ .

Question 5 · Modelled on HKDSE 2019 Paper 2 Q41 — the same three statements, applied to a specific obtuse-angled triangle

The coordinates of the points

A

B

and

C

are

(0, 0)

(6, 0)

and

(8, 4)

respectively. Which of the following are true?
I. The orthocentre of

△ A B C

lies outside

△ A B C

.
II. The circumcentre of

△ A B C

lies outside

△ A B C

.
III. The centroid of

△ A B C

lies inside

△ A B C

I and II only
I and III only
II and III only
I, II and III

Show answer

D. $∠ A B C$ is obtuse, so the orthocentre $(8, - 4)$ and circumcentre $(3, 4)$ are outside; a centroid is always inside.

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

Where this sits in the syllabus

In Paper 2 this topic usually appears around question 40 to 42, one question per paper. Most marks are lost by finding the wrong centre correctly: the centroid when the circumcentre was asked, or the other way round.

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

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