Informally, a quadrilateral is a figure with four straight sides, as in the first figure ABCD in the figure below. But consider the other six figures in the figure: the five plane figures NOPE, SILY, DART, CUTS, OPENS and the non-planar BENT. Should we call all these figures quadrilaterals? If you answer ‘no’ for any of them, how will you define a quadrilateral so that such a figure is excluded? As you can see, some care is needed to precisely define what we think of as a quadrilateral.
Step-by-step solution
Idea: Look at what goes wrong in each figure, then add one condition to the definition for each fault: no three vertices in a line, all four in one plane, sides meet only at their shared ends, and the figure closes up after exactly four sides.
- NOPE: E, P, O lie on one line, so sides OP and PE make one straight segment EO. The figure is really triangle NOE. Not a quadrilateral. Rule: no three vertices collinear.½ mark
- SILY: S, L, Y are collinear, so side YS runs back over side LY and the sides overlap. Not a quadrilateral. The same rule (no three collinear) removes it.½ mark
- BENT: T is lifted out of the plane of B, E, N, so the four sides do not lie in one plane. Not a (plane) quadrilateral; it is called a non-planar quadrilateral. Rule: all four points in one plane.½ mark
- CUTS: sides UT and SC cross each other at a point that is not a vertex. Usually not counted (it is a self-intersecting quadrilateral). Rule: every point other than the vertices lies on exactly one side.½ mark
- OPENS: it has five points and four segments OP, PE, EN, NS, and the last point S is not joined back to O. It does not close up. Not a quadrilateral. Rule: exactly four vertices, with sides AB, BC, CD and DA, so the last side returns to the first vertex.½ mark
- DART: four vertices in a plane, no three collinear, sides meet only at their ends, so it is a quadrilateral, a non-convex one (the angle at the dent D is more than 180°).½ mark
- Definition. Let A, B, C, D be four distinct points in a plane, no three of them collinear. The points on the segments AB, BC, CD and DA form quadrilateral ABCD if every such point other than A, B, C, D lies on exactly one of these four segments.1 mark
Answer to write in the exam
NOPE: E, P, O collinear ⇒ it is a triangle ⇒ not a quadrilateral
SILY: S, L, Y collinear ⇒ sides overlap ⇒ not a quadrilateral
BENT: vertices not in one plane ⇒ not a plane quadrilateral
CUTS: two sides cross ⇒ self-intersecting ⇒ excluded
OPENS: five points, not closed ⇒ not a quadrilateral
DART: a quadrilateral (non-convex)
Definition: A, B, C, D distinct points in a plane, no three collinear; the points on AB, BC, CD, DA form quadrilateral ABCD if every point other than A, B, C, D lies on exactly one of these segments.
Common mistakes that cost marks
- Calling NOPE a quadrilateral because it has four labelled points. With E, P, O in a line it is a triangle.
- Rejecting DART. A dent does not stop a figure from being a quadrilateral; it makes it non-convex.
- Writing a definition that forgets the plane (lets in BENT) or forgets the crossing rule (lets in CUTS).
How this can come in the exam
Four points P, Q, R, S lie in a plane with Q, R, S on one straight line (R between Q and S). The figure PQRS is
- a convex quadrilateral
- a non-convex quadrilateral
- a triangle, not a quadrilateral
- a self-intersecting quadrilateral
Show answer
(C) a triangle, not a quadrilateral
QR and RS lie on one line and make the single side QS, so the figure is triangle PQS. A quadrilateral needs no three vertices collinear.
Assertion (A): A four-sided figure whose two sides cross each other is not counted as an ordinary quadrilateral.
Reason (R): In a quadrilateral, every point other than the four vertices lies on exactly one side.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
Where two sides cross, that point lies on two sides, which the definition forbids. So R is true and explains A.
Try one yourself
Is a figure made of four points K, L, M, N with KL, LM, MN, NK drawn, where K, L, M, N do not lie in one plane, a quadrilateral in the usual sense? What is it called?
Show answer
No. A quadrilateral must lie in one plane. Such a figure is called a non-planar (skew) quadrilateral.
More questions like this
- 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?
- Can we similarly define a quadrilateral ABCD?
- 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?
- 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?
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