Two sides of a parallelogram are along the lines x+y=3 and x=y+3. If its diagonals intersect at (2, 4) , then one of its vertices is

Options

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

Answer

To solve the problem step by step, we need to find one of the vertices of the parallelogram given the equations of two sides and the intersection point of the diagonals.

### Step 1: Identify the lines
The two sides of the parallelogram are given by the equations:
1. \( x + y = 3 \) (Line 1)
2. \( x - y = 3 \) (Line 2)

### Step 2: Find the intersection of the two lines
To find the intersection point of these two lines, we can solve them simultaneously.

From Line 1:
\[ y = 3 - x \]

Substituting this into Line 2:
\[ x - (3 - x) = 3 \]
\[ x - 3 + x = 3 \]
\[ 2x - 3 = 3 \]
\[ 2x = 6 \]
\[ x = 3 \]

Now substituting \( x = 3 \) back into Line 1 to find \( y \):
\[ y = 3 - 3 = 0 \]

Thus, the intersection point is \( A(3, 0) \).

### Step 3: Use the midpoint of the diagonals
We know that the diagonals of a parallelogram bisect each other. The diagonals intersect at point \( O(2, 4) \). Since \( O \) is the midpoint of the diagonal \( AC \), we can use the midpoint formula.

Let \( C(x_C, y_C) \) be the coordinates of the vertex \( C \). The midpoint \( O \) can be expressed as:
\[ O = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) \]
Substituting the known values:
\[ (2, 4) = \left( \frac{3 + x_C}{2}, \frac{0 + y_C}{2} \right) \]

### Step 4: Set up equations from the midpoint
From the x-coordinates:
\[ 2 = \frac{3 + x_C}{2} \implies 4 = 3 + x_C \implies x_C = 1 \]

From the y-coordinates:
\[ 4 = \frac{0 + y_C}{2} \implies 8 = y_C \implies y_C = 8 \]

Thus, the coordinates of vertex \( C \) are \( C(1, 8) \).

### Step 5: Find the coordinates of the fourth vertex \( D \)
To find the coordinates of the fourth vertex \( D \), we can use the fact that the diagonals bisect each other. If \( B(x_B, y_B) \) is the coordinates of vertex \( B \), we can express the midpoint \( O \) as:
\[ O = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right) \]
Since we already know \( O(2, 4) \) and \( A(3, 0) \), we can find \( D \) using the relationship between the vertices.

### Step 6: Use the slopes to find vertex \( D \)
The slope of line \( AD \) should be equal to the slope of line \( BC \). We can find the slope of \( AD \) and set it equal to the slope of \( BC \) to find the coordinates of \( D \).

### Final Answer
After calculating, we find that one of the vertices of the parallelogram is \( C(1, 8) \).