To find the equation of the plane that passes through the midpoint of the line joining the points (1, 2, 3) and (3, 4, 5) and is perpendicular to the line segment, we will follow these steps:
### Step 1: Find the Midpoint of the Line Segment
The midpoint \( M \) of the line segment joining the points \( A(1, 2, 3) \) and \( B(3, 4, 5) \) can be calculated using the midpoint formula:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]
Substituting the coordinates of points A and B:
\[
M = \left( \frac{1 + 3}{2}, \frac{2 + 4}{2}, \frac{3 + 5}{2} \right) = \left( \frac{4}{2}, \frac{6}{2}, \frac{8}{2} \right) = (2, 3, 4)
\]
### Step 2: Find the Direction Ratios of the Line Segment
The direction ratios of the line segment joining the points \( A \) and \( B \) can be found by subtracting the coordinates of point A from point B:
\[
\text{Direction Ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) = (3 - 1, 4 - 2, 5 - 3) = (2, 2, 2)
\]
### Step 3: Write the Equation of the Plane
The general equation of a plane can be expressed as:
\[
a(x - x_0) + b(y - y_0) + c(z - z_0) = 0
\]
where \( (x_0, y_0, z_0) \) is a point on the plane and \( (a, b, c) \) are the direction ratios of the normal to the plane. Here, the point is the midpoint \( M(2, 3, 4) \) and the direction ratios are \( (2, 2, 2) \).
Substituting these values into the equation:
\[
2(x - 2) + 2(y - 3) + 2(z - 4) = 0
\]
Expanding this:
\[
2x - 4 + 2y - 6 + 2z - 8 = 0
\]
Combining like terms:
\[
2x + 2y + 2z - 18 = 0
\]
Thus, we can write:
\[
2x + 2y + 2z = 18
\]
### Step 4: Simplify the Equation
To simplify the equation, we can divide through by 2:
\[
x + y + z = 9
\]
### Final Answer
The equation of the plane is:
\[
\boxed{x + y + z = 9}
\]
