Definition of a quadrilateral · 2 marks

To prepare, let us first consider how we can define a triangle. Let A, B and C be three points. Can we say that ∆ABC consists of points on the three line segments AB, BC, CA?

Answer: Yes, provided A, B, C are not collinear. If the three points lie on one line, the three segments just make one line segment, not a triangle.

Step-by-step solution

Idea: A definition must rule out the case that breaks it. Three points on one line give overlapping segments, not a closed three-sided figure.

If A, B, C are not on one line, segments AB, BC and CA meet only at their ends and enclose a region. So ∆ABC is exactly the set of points on these three segments.1 mark
If A, B, C are on one line (say B between A and C), then AB and BC together make AC, and CA lies on top of them. We get only the segment AC: not a triangle. So the definition needs the condition “A, B, C non-collinear”.1 mark

Yes, as long as A, B and C are not collinear; then A, B, C are the vertices and AB, BC, CA the sides of ∆ABC.

Answer to write in the exam

A, B, C non-collinear ⇒ AB, BC, CA meet only at A, B, C ⇒ they form ∆ABC

A, B, C collinear ⇒ AB, BC, CA give only one line segment

∴ Yes, ∆ABC = points on AB, BC, CA, provided A, B, C are not collinear.

Common mistakes that cost marks

Answering a plain “yes” without the condition that the points are non-collinear.
Thinking three collinear points still give a “flat” triangle. They give a single segment with no inside.

How this can come in the exam

MCQ (1 mark)

Points X(1, 1), Y(2, 2) and Z(3, 3) are joined by XY, YZ and ZX. The figure formed is

an acute triangle
a right triangle
a line segment
an obtuse triangle

Show answer

(C) a line segment
X, Y, Z all lie on the line y = x, so the three segments make the single segment XZ.

Try one yourself

Three points P, Q, R are such that PQ = 3 cm, QR = 5 cm and PR = 8 cm. Do segments PQ, QR, RP form a triangle?