Question 1141617
<pre>
We start out with a square (which IS a rhombus for all sides are equal in
length. That's when the diagonals are equal in length, which, by the
Pythagorean theorem equal to {{{5*sqrt(2)}}}.
{{{c^2=a^2+b^2}}}
{{{c^2=25^2+25^2}}}
{{{c^2=25^2*2}}}
{{{c=sqrt(25^2*2)}}}
{{{c=5*sqrt(2)}}}

Then as we decrease the angle on the bottom left and increase the angle on
the bottom right, the green diagonal increases to 25+25 or 50, but never
quite gets to 50, for if it did, we'd only have a line segment 50 units
long.  The red diagonal shrinks to 0 but never quite gets to 0 for the same
reason.


{{{drawing(300,300,-.1,2.1,-.1,2.1,line(0,0,1,0),line(0,0,0,1) ,line(1,0,1,1),red(line(0,1,1,0)),
line(0,1,1,1),green(line(0,0,1,1)),
locate(.5,0,25),locate(1.05,.5,25) )}}}{{{drawing(300,300,-.1,2.1,-.1,2.1,
red(line(cos(1.2),sin(1.2),1,0)),
line(0,0,1,0),line(0,0,cos(1.2),sin(1.2)) ,line(1,0,1+cos(1.2),sin(1.2)),
line(cos(1.2),sin(1.2),1+cos(1.2),sin(1.2)),green(line(0,0,1+cos(1.2),sin(1.2))),
locate(.5,0,25),locate(1,.6,25) )}}}{{{drawing(300,300,-.1,2.1,-.1,2.1,line(0,0,1,0),line(0,0,cos(.5),sin(.5)) ,line(1,0,1+cos(.5),sin(.5)),red(line(cos(.5),sin(.5),1,0)),
line(cos(.5),sin(.5),1+cos(.5),sin(.5)),green(line(0,0,1+cos(.5),sin(.5))),
locate(.5,0,25),locate(1.44,.26,25) )}}}{{{drawing(300,300,-.1,2.1,-.1,2.1,line(0,0,1,0),line(0,0,cos(.25),sin(.25)) ,line(1,0,1+cos(.25),sin(.25)),red(line(cos(.25),sin(.25),1,0)),
line(cos(.25),sin(.25),1+cos(.25),sin(.25)),green(line(0,0,1+cos(.25),sin(.25))),
locate(.5,0,25), locate(1.4,.1,25) )}}}{{{drawing(300,300,-.1,2.1,-.1,2.1,
red(line(cos(.1),sin(.1),1,0)),
line(0,0,1,0),line(0,0,cos(.1),sin(.1)) ,line(1,0,1+cos(.1),sin(.1)),
line(cos(.1),sin(.1),1+cos(.1),sin(.1)),green(line(0,0,1+cos(.1),sin(.1))),
locate(.5,0,25),locate(1.5,.05,25) )}}}

Answer: the lengths of a diagonal can only be in the open interval from 0 to 50.  In interval notation that is (0,50) or 0 < x < 50.

Edwin</pre>