Which conditions should I check to see if a quadrilateral is a square?

[murshid_islam]

TL;DR

Conditions to check if a quadrilateral is a square

If I'm given the coordinates (edit: in a plane) of 4 vertices of a quadrilateral, which conditions should I check to see if it is a square? Will it be sufficient if I check if -

1. all 4 sides are equal
2. all 4 angles are right angles (or even 2 angles are right angles?)

or are there other conditions I need to check as well? Can I check any alternate set of conditions to verify if it's a square?

 

[jbriggs444]

TL;DR Summary: Conditions to check if a quadrilateral is a square

If I'm given the coordinates of 4 vertices of a quadrilateral, which conditions should I check to see if it is a square? Will it be sufficient if I check if -

1. all 4 sides are equal
2. all 4 angles are right angles (or even 2 angles are right angles?)

or are there other conditions I need to check as well? Can I check any alternate set of conditions to verify if it's a square?

These are coordinates in a plane? i.e. ordered pairs? Or coordinates in 3-space? i.e. ordered triples?

If you are working in the plane, all four sides equal plus one right angle means that it is a square.

In three space, one angle alone won't do it, but it seems clear that four would. However, a check that the cross products of all four pairs of adjacent sides are equal might be more convenient.

 

[murshid_islam]

These are coordinates in a plane? i.e. ordered pairs?

Yes, coordinates in a plane. Thank you. Edited my original post.

 

[murshid_islam]

If you are working in the plane, all four sides equal plus one right angle means that it is a square.

Any other alternate set of conditions I can check? Like checking if the opposite sides are parallel and the diagonals are equal?

 

[jbriggs444]

Any other alternate set of conditions I can check? Like checking if the opposite sides are parallel and the diagonals are equal?

With cartesian coordinates in the plane, compute the vector difference between each pair of adjacent endpoints. Two differences should be equal and opposite. The other two should be equal to the transpose of the first with one coordinate negated.

 

[Mark44]

Will it be sufficient if I check if -

1. all 4 sides are equal
2. all 4 angles are right angles (or even 2 angles are right angles?)

Assuming that both conditions are satisfied, these should be sufficient. If you have determined only that all four sides are equal, the figure could be a rhombus, sort of a diamond shape. If you also determine that two of the interior angles are equal, then that narrows things down to a specific kind of rhombus; namely a square.

 

[DaveC426913]

If you also determine that two of the interior angles are equal, then that narrows things down to a specific kind of rhombus; namely a square.

Wait. Correct me of I'm wrong but "Interior angles" in a closed polygon simply refers to any of the angles inside. (Here's a polygon with no parallels, to make the point):

I think you're thinking of this:

which is not a polygon, let alone a quadrilateral.


So to determine that a quadrilateral (a closed polygon) is square, you couldn't measure any two interior angles, since a quadrilateral has two pairs of equal interior angles. You'd have to measure adjacent angles.

Yes?

 

[jbriggs444]

So to determine that a quadrilateral (a closed polygon) is square, you couldn't measure any two interior angles, since a quadrilateral has two pairs of equal interior angles. You'd have to measure adjacent angles.

@Mark44 specifically mentioned a rhombus -- a four sided plane figure with four equal sides.

But yes, in a rhombus, the opposite angles are always equal and a test for squareness would be that two adjacent angles (and, therefore, all four angles) are equal.

 

[DaveC426913]

@Mark44 specifically mentioned a rhombus -- a four sided plane figure with four equal sides

Yes, but his criteria weren't sufficient to determine if its a square, which what i thought was his point.

 

[Mark44]

Yes, but his criteria weren't sufficient to determine if its a square, which what i thought was his point.

I misspoke a bit. I meant two interior right angles but wrote only equal interior angles, to elaborate on what the OP wrote. Sometimes the brain doesn't communicate well with the fingers.

 

[snorkack]

If you verify that 4 sides are equal, then it is sufficient to check that any one angle is right.
You don´t have to check that 4 angles are right, because if 3 angles are right, the fourth is. But checking that 2 angles are right is not enough because a trapezium has these. Indeed, that 3 (and therefore 4) angles are right is not enough because an oblong has these. You would need to check that 2 adjacent sides are equal, and 3 angles right. (2 adjacent sides equal and 2 angles right is, again, possible for a trapezium).

 

[Svein]

4 sides equal and both diagonals are equal.

 

[DaveC426913]

4 sides equal and both diagonals are equal.

How do you know the diagonals are equal if the diagonals are not in the diagram, let alone labelled such?

Is this a square?

 

[Mark44]

How do you know the diagonals are equal if the diagonals are not in the diagram, let alone labelled such?

Is this a square?
View attachment 359009

Just because the four sides are equal, it isn't necessarily a square. It could be a rhombus. That's the reason for checking to see if the diagonals are equal.

 

[snorkack]

Just because the four sides are equal, it isn't necessarily a square. It could be a rhombus. That's the reason for checking to see if the diagonals are equal.

And if two diagonals are equal, there is no point checking that four sides are equal. You need to check that two adjacent sides are equal.

 

[jbriggs444]

And if two diagonals are equal, there is no point checking that four sides are equal. You need to check that two adjacent sides are equal.

Then this is a square? Two equal adjacent sides and two equal diagonals.

 

[snorkack]

Then this is a square? Two equal adjacent sides and two equal diagonals.

View attachment 359150

Ouch! You´re right!
I was thinking of counterexample of oblong (two equal diagonals but the adjacent sides differ).
So how many sides need to be equal if two diagonals are? If three sides are equal and so are two diagonals, is the fourth side equal without checking?

 

[jbriggs444]

Ouch! You´re right!
I was thinking of counterexample of oblong (two equal diagonals but the adjacent sides differ).
So how many sides need to be equal if two diagonals are? If three sides are equal and so are two diagonals, is the fourth side equal without checking?