To find the slope of diagonal GE in a square, we need to consider the properties of the square's diagonals.
1. Identify the given slope: We are given the equation of the line on which diagonal FH lies:
[tex]\[
y - 3 = -\frac{1}{3}(x + 9)
\][/tex]
From this equation, it's clear that the slope of line FH is [tex]\( -\frac{1}{3} \)[/tex].
2. Understand the relationship between the diagonals: In a square, the diagonals are perpendicular to each other. The product of the slopes of two perpendicular lines is always [tex]\(-1\)[/tex].
3. Calculate the slope of GE: If the slope of FH is [tex]\( -\frac{1}{3} \)[/tex], let’s denote the slope of GE as [tex]\(m\)[/tex]. Since FH and GE are perpendicular,
[tex]\[
\left( -\frac{1}{3} \right) \cdot m = -1
\][/tex]
To find [tex]\(m\)[/tex], solve for [tex]\(m\)[/tex] by isolating it on one side of the equation:
[tex]\[
m = \frac{-1}{-\frac{1}{3}}
\][/tex]
Simplifying this results in
[tex]\[
m = 3
\][/tex]
Thus, the slope of diagonal GE is [tex]\(\boxed{3}\)[/tex].
