Can you come up with a definition that rules out self-intersecting quadrilaterals like CUTS?
Step-by-step solution
Idea: In CUTS two sides cross at a point that is not a vertex. That point belongs to two sides at once, so forbid exactly that.
- In CUTS, sides UT and SC cross at a point X that is not a vertex. X lies on two sides.1 mark
- In an ordinary quadrilateral, two sides meet only at a shared vertex. So we add the rule: every point of the figure other than the four vertices lies on exactly one side. This removes CUTS.1 mark
Answer to write in the exam
In CUTS, UT and SC cross at a non-vertex point lying on two sides
∴ Condition: every point other than the vertices lies on exactly one side.
Common mistakes that cost marks
- Saying “sides must not touch”: adjacent sides do touch, at their common vertex. The rule must allow that.
- Saying “the diagonals must not cross”: in a convex quadrilateral the diagonals do cross. The rule is about sides.
How this can come in the exam
Assertion (A): In quadrilateral ABCD, side AB and side BC have a common point.
Reason (R): In a quadrilateral, no point lies on two sides.
- 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
(C) A is true but R is false.
A is true: AB and BC share vertex B. R is false as stated: the vertices lie on two sides; the rule applies only to points other than the vertices.
Try one yourself
In a figure WXYZ, sides XY and ZW cross at a point O. Which condition of the definition of a quadrilateral fails?
Show answer
O is not a vertex but lies on two sides (XY and ZW), so “every point other than the vertices lies on exactly one side” fails: WXYZ is self-intersecting.
More questions like this
- 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.
- Let ABCD be a quadrilateral.
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