Key Concepts and Formulas

• Intersection of Lines: The coordinates of the intersection point of two lines satisfy the equations of both lines.
• Diagonals of a Parallelogram: The diagonals of a parallelogram bisect each other. The intersection point is the midpoint of each diagonal.
• Midpoint Formula: The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.
• Parallel Lines: Parallel lines have the same slope. A line parallel to $Ax + By + C = 0$ has the form $Ax + By + k = 0$.

Step-by-Step Solution

Step 1: Find the intersection point of the given lines, which represents a vertex of the parallelogram.

We are given two sides of the parallelogram as: $L_1: x + y = 3$ $L_2: x - y + 3 = 0 \implies x - y = -3$

The slopes of these lines are $-1$ and $1$, respectively. Since the slopes are different, the lines are not parallel and must represent adjacent sides. Let's call their intersection point vertex $A$. To find the coordinates of $A$, we solve the system of equations:

$x + y = 3 \quad \cdots (1)$ $x - y = -3 \quad \cdots (2)$

Adding equation (1) and equation (2): $(x+y) + (x-y) = 3 + (-3)$ $2x = 0$ $x = 0$

Substituting $x=0$ into equation (1): $0 + y = 3$ $y = 3$ Thus, vertex $A$ is $(0, 3)$.

Step 2: Find the vertex opposite to A (vertex C) using the midpoint property.

Let the intersection point of the diagonals be $M = (2, 4)$. Since the diagonals bisect each other, $M$ is the midpoint of $AC$. Let $C = (x_C, y_C)$. Using the midpoint formula:

$M = \left(\frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}\right)$ $(2, 4) = \left(\frac{0 + x_C}{2}, \frac{3 + y_C}{2}\right)$

Equating the x-coordinates: $2 = \frac{x_C}{2} \implies x_C = 4$

Equating the y-coordinates: $4 = \frac{3 + y_C}{2} \implies 8 = 3 + y_C \implies y_C = 5$

Therefore, vertex $C$ is $(4, 5)$.

Step 3: Determine the equations of the lines parallel to the given sides and passing through vertex C.

Since opposite sides of a parallelogram are parallel, the side opposite to $x + y = 3$ must be parallel to it and pass through $C(4, 5)$. Its equation is of the form $x + y = k_1$. Substituting the coordinates of $C$:

$4 + 5 = k_1 \implies k_1 = 9$ So, the equation of the line is $x + y = 9$.

Similarly, the side opposite to $x - y = -3$ must be parallel to it and pass through $C(4, 5)$. Its equation is of the form $x - y = k_2$. Substituting the coordinates of $C$:

$4 - 5 = k_2 \implies k_2 = -1$ So, the equation of the line is $x - y = -1$.

Step 4: Find the remaining vertices B and D.

Vertex $B$ is the intersection of the lines $x + y = 3$ and $x - y = -1$. Solving this system:

$x + y = 3 \quad \cdots (3)$ $x - y = -1 \quad \cdots (4)$

Adding (3) and (4): $2x = 2 \implies x = 1$ Substituting $x = 1$ into (3): $1 + y = 3 \implies y = 2$ So, vertex $B$ is $(1, 2)$.

Vertex $D$ is the intersection of the lines $x - y = -3$ and $x + y = 9$. Solving this system:

$x - y = -3 \quad \cdots (5)$ $x + y = 9 \quad \cdots (6)$

Adding (5) and (6): $2x = 6 \implies x = 3$ Substituting $x = 3$ into (6): $3 + y = 9 \implies y = 6$ So, vertex $D$ is $(3, 6)$.

Step 5: Check the options to see which vertex matches.

The vertices are $A(0, 3)$, $B(1, 2)$, $C(4, 5)$, and $D(3, 6)$. Comparing these with the given options:

(A) $(2, 1)$ (B) $(2, 6)$ (C) $(3, 5)$ (D) $(3, 6)$

Vertex $D(3, 6)$ matches option (D).

Common Mistakes & Tips

• Be careful with signs when solving systems of equations.
• Remember that the diagonals of a parallelogram bisect each other, meaning they divide each other into two equal parts.
• Always check your answer by verifying that all vertices satisfy the properties of a parallelogram.

Summary

We found the vertices of the parallelogram by using the properties of intersecting lines, parallel lines, and the midpoint formula. We determined the coordinates of all four vertices and then compared them to the provided options.

The final answer is $\boxed{(3, 6)}$, which corresponds to option (D).