To find the slope of diagonal GE, we can use properties of the diagonals of a square. In a square, the diagonals are perpendicular to each other and bisect each other.
1. First, identify the slope of the given diagonal, FH.
The equation of line [tex]\( FH \)[/tex] is given in point-slope form:
[tex]\[ y - 3 = -\frac{1}{3}(x + 9) \][/tex]
To convert this to slope-intercept form (y = mx + b), where [tex]\( m \)[/tex] is the slope:
[tex]\[ y - 3 = -\frac{1}{3}x - 3 \][/tex]
[tex]\[ y = -\frac{1}{3}x - 3 + 3 \][/tex]
[tex]\[ y = -\frac{1}{3}x \][/tex]
Hence, the slope of diagonal FH is [tex]\( -\frac{1}{3} \)[/tex].
2. Next, since the diagonals of a square are perpendicular to each other, the slope of the second diagonal, GE, must be the negative reciprocal of the slope of FH.
The negative reciprocal of [tex]\( -\frac{1}{3} \)[/tex] is calculated as follows:
[tex]\[ m_{FH} = -\frac{1}{3} \][/tex]
[tex]\[ m_{GE} = -\left(\frac{1}{-1/3}\right) = 3 \][/tex]
Therefore, the slope of diagonal GE is [tex]\( 3 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
