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?
Answer: Yes, DART is a quadrilateral (a non-convex one): it meets every condition of the definition. A quadrilateral is convex when all of its internal angles are less than 180°; DART is non-convex because its internal angle at D is more than 180°.
Step-by-step solution
Idea: Check DART against the definition of a quadrilateral, then look for the feature that the dent creates: an inside angle bigger than a straight angle.
- DART has four vertices in one plane, no three on a line, and its sides meet only at their shared ends. So it is a quadrilateral.½ mark
- At the dent D, the inside of the figure wraps round D by more than a straight angle: the internal angle ∠D is more than 180° (a reflex angle).½ mark
- In a quadrilateral without a dent, every internal angle is less than 180°. So define: a quadrilateral is convex if all its internal angles are less than 180°; otherwise it is non-convex.1 mark
DART is a quadrilateral, a non-convex one. A quadrilateral is convex when all its internal angles are less than 180°.
Answer to write in the exam
DART: 4 coplanar vertices, no three collinear, sides meet only at vertices ⇒ quadrilateral
Internal ∠D > 180° (reflex) at the dent
∴ Convex quadrilateral: all internal angles < 180°; DART is non-convex.
Common mistakes that cost marks
- Measuring the outside angle at D (less than 180°) and calling DART convex. The internal angle at the dent is the reflex one.
- Defining convex as “all angles acute”. A rectangle has right angles and is convex; the test is “less than 180°”.
How this can come in the exam
MCQ (1 mark)
A quadrilateral has internal angles 40°, 50°, 60° and 210°. It is
- convex
- non-convex
- self-intersecting
- impossible
Show answer
(B) non-convex
40 + 50 + 60 + 210 = 360, so it exists; the angle of 210° is more than 180°, so it is non-convex.
Try one yourself
Three internal angles of a quadrilateral are 35°, 45° and 50°. Is it convex?
Show answer
Fourth angle = 360° − 130° = 230° > 180°, so it is non-convex.
More questions like this
- 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.)
- In a quadrilateral ABCD, suppose AB ‖ DC. Can ABCD be non-convex? What if we instead assume AB = CD? What if we instead assume ∠A = ∠C?
- Consider three non-collinear points A, B, C and draw the lines AB, BC, CA. For every possible location of point D in the plane outside these lines, decide if ABCD is self-intersecting, non-convex, or convex. (Hint: the three lines divide the plane into 7 regions.)
All Quadrilaterals and parallelograms questions · All maths questions