Can you see how to rule out figures like OPENS?
Step-by-step solution
Idea: OPENS is a chain O–P–E–N–S of four segments whose ends O and S are not joined. A quadrilateral must be a closed figure with four vertices.
- OPENS uses five points O, P, E, N, S and four segments OP, PE, EN, NS. The end S is never joined back to the start O, so the figure is open.1 mark
- In the definition, start with exactly four distinct points A, B, C, D and use the four segments AB, BC, CD and DA. The side DA closes the figure, so an open chain like OPENS cannot fit.1 mark
Answer to write in the exam
OPENS: 5 points, segments OP, PE, EN, NS; S not joined to O ⇒ open figure
∴ Define with exactly 4 vertices A, B, C, D and sides AB, BC, CD, DA (closed).
Common mistakes that cost marks
- Counting segments only: OPENS does have four segments, but it has five vertices and is not closed.
- Thinking the fix is “no three collinear”: none of O, P, E, N, S need be collinear, yet OPENS is still not a quadrilateral.
How this can come in the exam
A figure is made of segments KL, LM, MN and NP, where K, L, M, N, P are five different points. Which is true?
- It is a convex quadrilateral
- It is a non-convex quadrilateral
- It is not a quadrilateral, because it is not closed
- It is a pentagon
Show answer
(C) It is not a quadrilateral, because it is not closed
There are only four segments and P is not joined to K, so the figure is an open chain. A pentagon would need a fifth side PK.
Try one yourself
Which side must be added to the open chain A–B–C–D (segments AB, BC, CD) to make quadrilateral ABCD?
Show answer
Side DA, joining the last vertex back to the first.
More questions like this
- 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.
- Let ABCD be a quadrilateral.
- You have used internal angles of quadrilaterals, but they too require an exact definition, just like how we gave one for a quadrilateral. Precisely define the internal angle of a quadrilateral at a given vertex. Your answer should work for a non-convex quadrilateral too. (Hint: use the opposite vertex as well.)
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