Problem: Suppose Q is a (convex) quadliateral with sides 5, 7, 8 and 9. Suppose also that the spread of the diagonals is 3/4. Find the diagonals of the quadrilateral. Generalize.WildTrig Video 64
Solution: As it turns out there is no convex quadrilateral with the given sides and and a diagonal spread of 3/4. The minimum diagonal spread of a quadrilateral with sides 5,7,8,9 is approximately .7709, representing an acute angle of approximately 61.04 degrees. By changing the 5 to a 6, we can surmise from the Sage interact below that there is a unique quadrilateral with diagonal spread 3/4, although we don't know the exact quadrilateral.
A non convex quadrilateral with sides 5,7,8,9 and diagonal spread of 3/4.
If the quadrilateral is allowed to be non convex, then there are two solutions. The diagonals and are shown in red.
Question: Find the quadrilaterals with rational diagonal spread whose sides have integer length.
There are some easy examples -- Any kite with sides of integer length. Also, any quadrilateral with integer sides can be flexed to an infinite number of positions so that the diagonal spread is rational. Note: Partial solution.
A generalization of problem 4: Given the consecutive sides of a quadrilateral and the spread of the diagonals, determine the quadrilateral.Discussion: As is remarked in the lecture, it is easy to prove that all you need is the 4 sides and 1 diagonal to completely determine the quadrilateral. You can also get by with the 4 sides and the angle between two adjacent sides. But, in general, four sides and the spread of the diagonals do not suffice, as you can see by experimenting with the Sage interact below.
Setup: Label the vertices of the quadrilateral in counterclock order with at the bottom, on the right, at the top and on the left. By relabeling, we can assume that and . By scaling, normalize the quadrilateral so that . Set up a coordinate system so that and and and lie above or on the -axis.
We can imagine the quadrilateral flexing as moves along a circle of radius about , starting at a triangle with sides of length , (if ), and or a segment of length (if ) and ending at a triangle with sides , , and (if ) or sides , , and (if ) or a segment of length (if ). Note that $a\lt b+c+dThis motion is controlled by the position of , the first coordinate of , for we can express in terms of : , where , , and , and . We can compute the leftmost and rightmost positions of , call them and respectively. In the interact, the variable sets and computes the quadrilateral.