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.)
Step-by-step solution
Idea: The name tells us the sides: consecutive letters are joined, and the last letter is joined to the first. So two names describe the same quadrilateral exactly when they give the same four sides.
- ABCD has sides AB, BC, CD, DA. Going round the boundary from any starting vertex in the same direction gives ABCD, BCDA, CDAB, DABC: 4 names.½ mark
- Going round the other way gives DCBA, ADCB, BADC, CBAD: 4 more. Each still has the sides AB, BC, CD, DA. Total 8 names.½ mark
- There are 4 × 3 × 2 × 1 = 24 orders of four letters. The other 16 put two opposite vertices next to each other. For example ACBD has sides AC, CB, BD, DA: AC and BD are the diagonals of ABCD, and the figure ACBD has crossing sides (see the picture). So it is a different figure, not ABCD.1 mark
Check: Count: 24 orders in all. Each cyclic order round the boundary can be read from 4 starting points in 2 directions, so 24 ÷ 8 = 3 different figures can be made from four points; ABCD is one of them.
Answer to write in the exam
Sides of ABCD: AB, BC, CD, DA (consecutive letters joined, last to first)
Same direction: ABCD, BCDA, CDAB, DABC; opposite direction: DCBA, ADCB, BADC, CBAD
Any other order, e.g. ACBD, has sides AC, CB, BD, DA ⇒ uses a diagonal ⇒ a different figure
∴ Exactly 8 names (4 starting vertices × 2 directions).
Common mistakes that cost marks
- Thinking any arrangement of the letters names the same quadrilateral, e.g. writing ABDC for ABCD.
- Forgetting the reverse direction and listing only 4 names.
- Forgetting that the last letter is joined to the first, so BCDA has side AB.
How this can come in the exam
Which of the following names the same quadrilateral as PQRS?
- PRQS
- QPRS
- SRQP
- PQSR
Show answer
(C) SRQP
SRQP goes round the boundary in the reverse direction and keeps the sides SR, RQ, QP, PS. PRQS, QPRS and PQSR each put two opposite vertices next to each other.
Try one yourself
Write all the names of quadrilateral WXYZ.
Show answer
WXYZ, XYZW, YZWX, ZWXY, ZYXW, WZYX, XWZY, YXWZ (8 names).
More questions like this
- 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.)
- 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?
All Quadrilaterals and parallelograms questions · All maths questions