Why do ionic compounds conduct electric current when they are melted or dissolved in water?

It is beacuse the ions in the melted or aqueous ionic compound is mobile and can freely move through the fluid and conduct electricity.

The complete question in the attached figure

we have that

Part A) . Name all pairs of similar triangles in the diagram

the answer is
triangle ABG
triangle ACF
triangle ADE

Part B)  Find the lengths of segment BG
we know that
AB/BG=AD/DE----------> BG=(AB*DE)/AD
DE=6 ft
BD=6.32 ft
AB=BC=CD=6.32/2----------> 3.16 ft
AD=6.32+3.16------> AD=9.48 ft
BG=(AB*DE)/AD---------> 3.16*6/9.48----------> BG=2 ft

the answer Part B) is 
BG=2 ft

Part C)  Find the lengths of segment CF
we know that
AC/CF=AD/DE----------> CF=(AC*DE)/AD
AC=6.32 ft
DE=6 ft
AD=9.48 ft
CF=(AC*DE)/AD---------> 6.32*6/9.48----------> CF=4 ft

the answer Part C) is 
CF=4 ft