I was just looking at what Heureka was saying. It took me a while to work it out.

B
This is the triangle that Heureka was describing
The radius that bisects the chord is perpendicular to the chord so <AHE=90 degrees
which means <AEH=30 degrees (Angle sum of a triangle)
Using trig I know that AH=0.5*AE=2.25 but I am not allowed to use trig. 
Now Heureka says that AH=HB
Edited: that is because AB is one radius and AH is 1/2 radius so HB must also be a 1.2 radius
Join EB forming a new triangle HEB
Now triangle HEB is congruent to triangle HEA because all the angles are congruent and they have a common corresponding side.
therefore triangle AEB is equilateral and HA=HB
therefore all the sides are 4.5 inches
therefore HA=HB=2.25 inches
$$\\x^2=4.5^2-2.25^2\\
x^2=20.25-5.0625\\
x^2=15.1875\\
x^2=151875/10000\\
x=\sqrt{151875/10000}\\
x=\sqrt{25*25*81*3/10000}\\
x=(25*9/100)\sqrt{3}\\
x=(2.25)\sqrt{3}\\
side = 4.5\sqrt{3}\;\;inches$$
Thanks Heureka