Can we similarly define a quadrilateral ABCD?
Step-by-step solution
Idea: Copy the triangle definition (“the points on the segments joining the vertices in order”) and then ask what extra can go wrong with four points that could not happen with three.
- Try the same idea: ABCD is the set of points on AB, BC, CD and DA. With three points the only failure was “collinear”. With four points new failures appear.½ mark
- Failure 1: three of the points may be collinear. Then two sides lie along one line, giving a triangle or overlapping sides. Condition: no three of A, B, C, D collinear.½ mark
- Failure 2: four points need not lie in one plane (three points always do). Condition: A, B, C, D lie in one plane.1 mark
- Failure 3: two sides may cross, like a figure-of-eight. Condition: every point other than A, B, C, D lies on exactly one of the four segments.½ mark
- So: A, B, C, D distinct points in a plane, no three collinear, and 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.½ mark
Answer to write in the exam
Triangle: only need A, B, C non-collinear
Quadrilateral: (1) no three of A, B, C, D collinear
(2) A, B, C, D in one plane
(3) every point other than A, B, C, D on exactly one of AB, BC, CD, DA
∴ Similar definition works only with these three conditions added.
Common mistakes that cost marks
- Thinking “the four points are not all on one line” is enough. Even then three of them may be collinear, or two sides may cross.
- Forgetting the plane condition: three points always lie in a plane, but four points need not.
How this can come in the exam
Why is “three non-collinear points” enough to define a triangle, but “four points, not all collinear” is not enough to define a quadrilateral? Give one example.
Show answer
Any three non-collinear points lie in one plane and their three segments can only meet at the ends, so they always form a triangle (1 mark). With four points, three may still be collinear, the points may not be coplanar, or two sides may cross; for example A(0, 0), B(2, 2), C(2, 0), D(0, 2) joined in order gives crossing sides AB and CD (1 mark).Try one yourself
Points A(0, 0), B(4, 0), C(0, 3), D(4, 3) are joined in the order AB, BC, CD, DA. Is ABCD a quadrilateral?
Show answer
No. Sides BC and DA cross at (2, 1.5), which lies on two sides. ABCD is self-intersecting. (Joined as A, B, D, C it would be a rectangle.)
More questions like this
- 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?
- 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?
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