To solve this problem carefully, let's analyze the given information step by step.
1. Identify the Equation of Diagonal GE:
The line equation for diagonal GE is provided as [tex]\( y - 3 = 3(x + 4) \)[/tex]. We need to transform this equation into slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope.
Start by expanding and simplifying:
[tex]\[
y - 3 = 3(x + 4)
\][/tex]
[tex]\[
y - 3 = 3x + 12
\][/tex]
[tex]\[
y = 3x + 15
\][/tex]
From this final equation, we can see that the slope [tex]\( m \)[/tex] of diagonal GE is [tex]\( 3 \)[/tex].
2. Understand the Properties of a Square:
In a square, the diagonals are perpendicular bisectors of each other. This means that if two lines are perpendicular, the product of their slopes is [tex]\(-1\)[/tex].
3. Calculate the Slope of Diagonal FH:
If diagonal GE has a slope of [tex]\( 3 \)[/tex], the slope of diagonal FH must satisfy the condition of perpendicularity:
[tex]\[
(\text{slope of GE}) \times (\text{slope of FH}) = -1
\][/tex]
[tex]\[
3 \times (\text{slope of FH}) = -1
\][/tex]
Solving for the slope of FH:
[tex]\[
\text{slope of FH} = \frac{-1}{3}
\][/tex]
4. Verify the Correct Answer:
Based on our calculations, the slope of diagonal FH is [tex]\( -\frac{1}{3} \)[/tex].
In conclusion, the correct answer for the slope of diagonal FH is:
[tex]\[
-\frac{1}{3}
\][/tex]
