The problem asks for the equation of the diagonal BD of a parallelogram ABCD, given the coordinates of three consecutive vertices: A(1, 3), B(-1, 2), and C(3, 5).

In a parallelogram, the diagonals bisect each other. This means the midpoint of diagonal AC is the same as the midpoint of diagonal BD.

Finding the Midpoint of AC

We use the midpoint formula \( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \). The coordinates of A are (1, 3) and C are (3, 5).

Let M be the midpoint of AC.

The x-coordinate of M is \( \frac{1 + 3}{2} = \frac{4}{2} = 2 \).

The y-coordinate of M is \( \frac{3 + 5}{2} = \frac{8}{2} = 4 \).

So, the midpoint of AC is M(2, 4).

Finding the Coordinates of Vertex D

Since the diagonals bisect each other, the midpoint of BD is also M(2, 4). We know the coordinates of B are (-1, 2). Let the coordinates of the unknown vertex D be (x, y).

Using the midpoint formula for BD:

\( M = \left(\frac{x_B + x_D}{2}, \frac{y_B + y_D}{2}\right) \)

\( (2, 4) = \left(\frac{-1 + x}{2}, \frac{2 + y}{2}\right) \)

Equating the x-coordinates:

\( 2 = \frac{-1 + x}{2} \)

\( 4 = -1 + x \)

\( x = 4 + 1 \)

\( x = 5 \)

Equating the y-coordinates:

\( 4 = \frac{2 + y}{2} \)

\( 8 = 2 + y \)

\( y = 8 - 2 \)

\( y = 6 \)

Therefore, the coordinates of vertex D are (5, 6).

Finding the Equation of the Diagonal BD

Now we need to find the equation of the line passing through B(-1, 2) and D(5, 6). We can use the two-point form of the equation of a line: \( \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} \).

Let \((x_1, y_1) = (-1, 2)\) and \((x_2, y_2) = (5, 6)\).

Substitute these values into the formula:

\( \frac{y - 2}{x - (-1)} = \frac{6 - 2}{5 - (-1)} \)

\( \frac{y - 2}{x + 1} = \frac{4}{6} \)

Simplify the fraction on the right side:

\( \frac{y - 2}{x + 1} = \frac{2}{3} \)

Now, cross-multiply:

\( 3(y - 2) = 2(x + 1) \)

Distribute the numbers:

\( 3y - 6 = 2x + 2 \)

Rearrange the terms to get the equation in the form Ax + By + C = 0:

\( 0 = 2x - 3y + 2 + 6 \)

\( 2x - 3y + 8 = 0 \)

This is the equation of the diagonal BD.

Verification with Options

We compare our derived equation \( 2x - 3y + 8 = 0 \) with the given options:

1. 2x - 3y + 2 = 0
2. 3x - 2y + 5 = 0
3. 2x - 3y + 8 = 0
4. 3x - 2y - 5 = 0

Our equation matches option 3.

The final answer is \( 2x - 3y + 8 = 0 \).