Certainly! Let's solve the problem step-by-step.
We are given three points:
- [tex]\( E(-1, d) \)[/tex]
- [tex]\( F(1, 1) \)[/tex]
- [tex]\( G(3, -5) \)[/tex]
It is also given that point [tex]\( F \)[/tex] is the midpoint of the line segment [tex]\( EG \)[/tex].
### Step 1: Using the Midpoint Formula
The coordinates of the midpoint [tex]\( M \)[/tex] of a line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Here, point [tex]\( F \)[/tex] is the midpoint of [tex]\( E \)[/tex] and [tex]\( G \)[/tex]. Therefore:
[tex]\[ F = \left( \frac{-1 + 3}{2}, \frac{d + (-5)}{2} \right) \][/tex]
Given that [tex]\( F(1, 1) \)[/tex], we can now set up the equations as follows:
### Step 2: Set up the equations for the x-coordinates
[tex]\[ 1 = \frac{-1 + 3}{2} \][/tex]
### Step 3: Solve for the x-coordinates to verify
[tex]\[ 1 = \frac{2}{2} \][/tex]
[tex]\[ 1 = 1 \][/tex]
The x-coordinate part is verified indeed. Now, let's move on to the y-coordinates.
### Step 4: Set up the equations for the y-coordinates
[tex]\[ 1 = \frac{d + (-5)}{2} \][/tex]
### Step 5: Solve for [tex]\( d \)[/tex]
[tex]\[ 1 = \frac{d - 5}{2} \][/tex]
[tex]\[ 2 \times 1 = d - 5 \][/tex]
[tex]\[ 2 = d - 5 \][/tex]
Add 5 to both sides of the equation:
[tex]\[ 2 + 5 = d \][/tex]
[tex]\[ d = 7 \][/tex]
### Conclusion
The value of [tex]\( d \)[/tex] is [tex]\( \boxed{7} \)[/tex].
