Take any 4 distinct non-collinear points A, B, C and D. Will segments AB, BC, CD and DA always form a quadrilateral as we visualise it?
Step-by-step solution
Idea: “Non-collinear” only says the four points are not all on one line. Give one example for each way the segments can fail to look like a quadrilateral.
- Three collinear: A(0, 0), B(2, 0), C(4, 0), D(2, 3). AB and BC make one segment AC, so we get triangle ACD.½ mark
- Crossing sides: A(0, 0), B(4, 3), C(4, 0), D(0, 3). AB and CD cross at (2, 1.5), so the figure is self-intersecting.½ mark
- Not in one plane: three corners of a box on the floor and the fourth point on the ceiling. The segments form a bent, non-planar figure.½ mark
- So the answer is no; extra conditions are needed to get a quadrilateral as we picture it.½ mark
Check: Crossing point in the second example: AB is y = 0.75x, CD is y = 3 − 0.75x; they meet where 1.5x = 3, x = 2, y = 1.5 ✓, inside both segments.
Answer to write in the exam
A(0, 0), B(2, 0), C(4, 0), D(2, 3): A, B, C collinear ⇒ triangle ACD
A(0, 0), B(4, 3), C(4, 0), D(0, 3): AB and CD cross ⇒ self-intersecting
Four points not in one plane ⇒ non-planar figure
∴ No, the segments need not form a quadrilateral.
Common mistakes that cost marks
- Reading “non-collinear” as “no three collinear”. The first only rules out all four on one line.
- Drawing only nice examples and concluding “always yes”. One counterexample is enough to answer no.
How this can come in the exam
Points P(0, 0), Q(6, 0), R(0, 4), S(6, 4) are joined in the order PQ, QR, RS, SP. The figure is
- a rectangle
- a parallelogram that is not a rectangle
- self-intersecting
- a triangle
Show answer
(C) self-intersecting
QR and SP cross at (3, 2), a point on two sides, so PQRS is self-intersecting. (P, Q, S, R in that order would give a rectangle.)
Try one yourself
Give four points, not all on one line, which when joined in order give a triangle rather than a quadrilateral.
Show answer
For example A(0, 0), B(1, 1), C(2, 2), D(3, 0): A, B, C are collinear, so the figure is triangle ACD.
More questions like this
- Can you come up with a definition that rules out self-intersecting quadrilaterals like CUTS?
- Can you see how to rule out figures like OPENS?
- Note that quadrilateral ABCD can also be denoted as BCDA, CDAB, DABC, DCBA, ADCB, BADC or CBAD but not by any other sequence of vertices. (Draw ABCD and trace the vertices in various orders to see why.)
- Should DART in the figure be considered a quadrilateral? It seems different from our usual mental picture of a quadrilateral because it has a dent, as if someone has taken a bite out of triangle ART! We call DART a non-convex quadrilateral. A convex quadrilateral is one without a dent. How can we define this precisely?
- There are other ways of testing convexity. For example, visually verify that the diagonals of a convex quadrilateral intersect while those of a non-convex quadrilateral don’t intersect.
All Quadrilaterals and parallelograms questions · All maths questions