If four points `(x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3))` and `(x_(4),y_(4))` taken in order in a parallelogram, then:

Options

A. `x_(1)-x_(2)+x_(3)-x_(4)=0`
B. `y_(1)-y_(2)+y_(3)+y_(4)=0`
C. `x_(1)+x_(2)-x_(3)-x_(4)=0`
D. `y_(1)+y_(2)-y_(3)-y_(4)=0`

Answer

To determine the relationship between the coordinates of the points in a parallelogram, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**: We have four points of a parallelogram given as \( (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4) \).

2. **Understanding Midpoints**: In a parallelogram, the midpoints of the diagonals are the same. The diagonals are formed by connecting opposite vertices. Here, the diagonals are \( AC \) and \( BD \).

3. **Calculate the Midpoint of Diagonal AC**:
- The midpoint \( M_{AC} \) of diagonal \( AC \) (connecting points \( (x_1, y_1) \) and \( (x_3, y_3) \)) can be calculated using the midpoint formula:
\[ M_{AC} = \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) \]

4. **Calculate the Midpoint of Diagonal BD**:
- The midpoint \( M_{BD} \) of diagonal \( BD \) (connecting points \( (x_2, y_2) \) and \( (x_4, y_4) \)) is:
\[ M_{BD} = \left( \frac{x_2 + x_4}{2}, \frac{y_2 + y_4}{2} \right) \]

5. **Set the Midpoints Equal**: Since the midpoints of the diagonals are equal in a parallelogram, we set \( M_{AC} = M_{BD} \):
\[ \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) = \left( \frac{x_2 + x_4}{2}, \frac{y_2 + y_4}{2} \right) \]

6. **Equate the Corresponding Coordinates**:
- For the x-coordinates:
\[ \frac{x_1 + x_3}{2} = \frac{x_2 + x_4}{2} \]
- Multiplying through by 2 gives:
\[ x_1 + x_3 = x_2 + x_4 \quad \text{(1)} \]

- For the y-coordinates:
\[ \frac{y_1 + y_3}{2} = \frac{y_2 + y_4}{2} \]
- Multiplying through by 2 gives:
\[ y_1 + y_3 = y_2 + y_4 \quad \text{(2)} \]

7. **Rearranging the Equations**:
- From equation (1):
\[ x_1 + x_3 - x_2 - x_4 = 0 \quad \text{or} \quad x_1 + x_3 = x_2 + x_4 \]

- From equation (2):
\[ y_1 + y_3 - y_2 - y_4 = 0 \quad \text{or} \quad y_1 + y_3 = y_2 + y_4 \]

### Final Result:
Thus, the conditions for the points to form a parallelogram are:
1. \( x_1 + x_3 = x_2 + x_4 \)
2. \( y_1 + y_3 = y_2 + y_4 \)