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Centres of Triangles in the Coordinate Plane: Paper 2 Challenge

Challenge tier. Each question is built around a trap: a condition with two solutions, a distractor that is the right number for the wrong centre, or a statement that holds only some of the time. Work the standard Paper 2 set first. Answers sit under each question. Detailed solutions can be requested by email.

5 questions · about 12 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 · Challenge — a trap: a circumcentre on a side means a right angle, and the condition has two solutions

The coordinates of the points

A

and

B

are

(- 2, 1)

and

(6, 7)

respectively.

C

is a point on the

x

-axis such that the circumcentre of

△ A B C

lies on

A B

. Find the

x

-coordinate of

C

$5$ only
$- 1$ only
$1$ or $- 5$
$- 1$ or $5$

Show answer

D. $∠ A C B = 9 0^{\circ}$ , so $(x + 2) (x - 6) + 7 = 0$ , giving $x = - 1$ or $5$ . Choosing one root only is the trap.

Question 2 · Challenge — modelled on HKDSE 2017 Paper 2 Q41, with the other centres planted as distractors

The equations of the three sides of a triangle are

x = 0

y = 0

and

3 x + 4 y = 12 k

, where

k

is a positive constant. If the in-centre of the triangle lies on the straight line

x + 2 y = 9

, then

k =

$\frac{9}{5}$
$\frac{27}{10}$
$3$
$\frac{9}{2}$

Show answer

C. The sides are $4 k$ , $3 k$ , $5 k$ , so the in-centre is $(k, k)$ and $3 k = 9$ . The first two options use the circumcentre and the centroid.

Question 3 · Challenge — runs HKDSE 2021 Paper 2 Q41 backwards with no vertex at the origin

The coordinates of the points

A

and

B

are

(- 2, - 1)

and

(6, 3)

respectively. If the orthocentre of

△ A B C

(3, 4)

, then the coordinates of

C

are

$(1, 8)$
$(5, 10)$
$(- 3, 6)$
$(5, 6)$

Show answer

A. $C$ lies on $2 x + y - 10 = 0$ (perpendicular to $A B$ through $H$ ) and on $x + y - 9 = 0$ (perpendicular to $A H$ through $B$ ).

Question 4 · Challenge — modelled on HKDSE 2024 Paper 2 Q41, with the triangle given in letters so every statement must hold in general

The coordinates of the points

P

Q

and

R

are

(0, a)

(- b, 0)

and

(b, 0)

respectively, where

a

and

b

are positive numbers. Which of the following must be true?
I. The centroid, the orthocentre, the in-centre and the circumcentre of

△ P QR

all lie on the

y

-axis.
II. The circumcentre of

△ P QR

lies inside

△ P QR

.
III. If

a > b

, then the orthocentre of

△ P QR

lies inside

△ P QR

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

Show answer

C. II fails when $a < b$ , since $∠ QP R$ is then obtuse. If $a > b$ , every angle is acute.

Question 5 · Challenge — a technique the standard set never uses: the centroid of a triangle with one moving vertex

The coordinates of the points

A

and

B

are

(1, 2)

and

(7, 4)

respectively.

C

is a point moving on the straight line

x - y + 1 = 0

. As

C

moves, the centroid of

△ A B C

moves along a straight line. The equation of that straight line is

$x - y + 1 = 0$
$3 x - 3 y - 1 = 0$
$3 x - 3 y + 1 = 0$
$x - y - 1 = 0$

Show answer

B. With $C = (t, t + 1)$ , the centroid is $(\frac{t + 8}{3}, \frac{t + 7}{3})$ , so $x - y = \frac{1}{3}$ .

Continue practising this topic: Try the Paper 2 multiple-choice to build the basics, or move to Paper 1 long questions — also available as Paper 1 Challenge.

Where this sits in the syllabus

The hardest Paper 2 questions on this topic give a centre and ask for a vertex, or ask which statements must be true for every triangle of a given shape. A quick sketch, with the obtuse angle marked if there is one, settles most of them.

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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