Can a quadrilateral be both self-intersecting and non-planar?
Step-by-step solution
Idea: Self-intersecting means two opposite sides cross. Use the fact that two lines that meet always lie in a single plane.
- Suppose ABCD is self-intersecting. Adjacent sides already meet at a vertex, so the crossing must be between opposite sides, say AB and CD, at a point E.½ mark
- Lines AB and CD meet at E. Two lines that meet determine exactly one plane, which contains both lines.1 mark
- A, B lie on line AB and C, D lie on line CD, so all four vertices lie in that plane. ABCD is planar. So a quadrilateral cannot be both self-intersecting and non-planar.½ mark
Answer to write in the exam
Let opposite sides AB and CD of ABCD cross at E
Lines AB and CD intersect at E ⇒ they lie in one plane
A, B ∈ line AB and C, D ∈ line CD ⇒ A, B, C, D lie in that plane
∴ ABCD is planar; a quadrilateral cannot be both self-intersecting and non-planar.
Common mistakes that cost marks
- Saying “yes” by picturing two sticks that pass over each other in space. Passing over is not crossing: they do not meet.
- Forgetting to say why the crossing must be between opposite sides.
How this can come in the exam
Assertion (A): If sides PQ and RS of a four-sided figure PQRS meet at a point, then P, Q, R, S lie in one plane.
Reason (R): Two intersecting lines lie in exactly one plane.
- 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.
Lines PQ and RS meet, so they lie in one plane, which contains P, Q, R and S. R explains A.
Try one yourself
The diagonals AC and BD of a four-sided figure ABCD in space intersect. Must ABCD be planar?
Show answer
Yes. Lines AC and BD meet, so they lie in one plane, and that plane contains A, B, C and D.
More questions like this
- To test if a given quadrilateral is a parallelogram, do we have to check that the opposite sides are parallel? Are there other ways to test this?
- Recall the following properties of a parallelogram that we proved last year.
- If the converse of any of the three properties is true, it can be used as an alternate way to show that a quadrilateral is a parallelogram. To explore this, let us write the converses. Can you experiment and guess what the answers are?
- If the opposite sides of a quadrilateral are of equal length, then it is a parallelogram.
- If the opposite angles of a quadrilateral are equal, then it is a parallelogram.
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