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.
Step-by-step solution
Idea: Draw both kinds and join opposite vertices. In a convex quadrilateral each diagonal splits it into two triangles and the diagonals cross. At a dent, one diagonal goes outside the figure.
- Convex ABCD: draw AC and BD. They cross at a point inside ABCD. Try several convex shapes (square, kite, trapezium): the diagonals always cross.1 mark
- Non-convex DART: the dent D lies inside triangle ART. Diagonal DR runs inside the figure, but diagonal AT runs outside it, on the far side of D. The segments AT and DR do not meet (the line DR would reach AT only if extended beyond D).1 mark
Answer to write in the exam
Convex ABCD: diagonals AC and BD intersect at a point inside ABCD
Non-convex DART: D inside ∆ART; diagonal AT lies outside the figure, DR inside
∴ Segments AT and DR do not intersect. Convex ⇔ diagonals intersect.
Common mistakes that cost marks
- Extending the diagonals into full lines. The lines AT and DR do meet; the test is about the diagonal segments.
- Drawing only the inside diagonal of DART and forgetting the one that goes outside.
How this can come in the exam
Assertion (A): In a non-convex quadrilateral, one diagonal lies outside the quadrilateral.
Reason (R): A non-convex quadrilateral has an internal angle greater than 180°.
- 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
(A) Both A and R are true, and R is the correct explanation of A.
Both are true. The reflex angle at the dent pushes that vertex inside the triangle of the other three, so the diagonal joining its two neighbours passes outside. R explains A.
Try one yourself
In quadrilateral WXYZ, the diagonals WY and XZ meet at a point O inside the figure. Is WXYZ convex?
Show answer
Yes. Diagonals that intersect mean the quadrilateral is convex.
More questions like this
- 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.)
- Can a quadrilateral be both self-intersecting and non-planar?
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