In a triangle, with sides say $a,b,c$ we know that $a+b\geq{c}$ and $|a-b|\leq{c}$.
What are the inequalities we can form given the sides of the quadrilateral say $a,b,c,d$ where these are unknown to us.
Specifically I was doing the following problem when this question came to my mind.
In a trapezium, the lengths of the two parallel sides are $6$ and $10$ units. If one of the oblique sides has length $1$ unit, then prove that the length of the other oblique side is greater than $3$ units but less than $5$ units.
I solved this problem by dropping perpendiculars from the vertices of the smaller parallel side to the bigger one and then applying Pythagoras Theorem to get the length of the other oblique side. I was wondering of there is a simpler approach to this problem and in general what all can we say about the lengths of sides that form a quadrilateral.
[Math] Sides of a quadrilateral
geometric-inequalitiesgeometrypolygonsquadrilateral
Related Solutions
As stated the problem can have many solutions.
For example:
Let the quadrilateral $ABCD$ such that $AD=DB=BC=x$ and $AC=\sqrt{3}x$. See figure 1.
Figure 1
The angle $b$ depends on $a$ as the expression:
$$b = \frac{a}{2}- \frac {\pi}{2}+ \arccos(- \frac{1}{2 \sin {\frac{a}{2}}}+ \sin {\frac{a}{2}})$$
So we have many solutions, for example: $a=\frac{\pi}{3}$ and $b=\frac{\pi}{3}$ or $a=\frac{\pi}{4}$ and $b=\frac{\pi}{2}$ as shown in figure 2.
Figure 2
Quadrilaterals are not rigid: you can fix a side and move one of the other vertices freely while respecting all side lengths. The area changes when you move the vertex.
(image from http://www.mathsisfun.com/definitions/rigid.html)
Best Answer
You can, in fact generalize the triangle inequality to any other polygon. Try to prove by induction that for all polygons with number of sides $n \ge 3$ that the length of any side is always less than the sum of the other side lengths.
For quadrilaterals, Euler's theorem states that in any convex quadrilateral, $a^2+b^2+c^2+d^2 \ge p^2 + q^2$ where p and q are the diagonals of the quadrilateral. You can prove this by making some constructions and using the law of cosines.
There are specific inequalities for special quadrilaterals(cyclic, parallelograms, squares etc) but these are the only two general inequalities I can think of.