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 solve the problem of finding the relationship between the coordinates of the vertices of a parallelogram, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**: Let the points of the parallelogram be \( A(x_1, y_1) \), \( B(x_2, y_2) \), \( C(x_3, y_3) \), and \( D(x_4, y_4) \).

2. **Find the Midpoints of the Diagonals**:
- The midpoint \( O \) of diagonal \( AC \) can be calculated using the midpoint formula:
\[ O = \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) \]
- Similarly, the midpoint \( O \) of diagonal \( BD \) is:
\[ O = \left( \frac{x_2 + x_4}{2}, \frac{y_2 + y_4}{2} \right) \]

3. **Set the Midpoints Equal**: Since both expressions represent the same point \( O \), we can set the x-coordinates and y-coordinates equal to each other:
- For the x-coordinates:
\[ \frac{x_1 + x_3}{2} = \frac{x_2 + x_4}{2} \]
- For the y-coordinates:
\[ \frac{y_1 + y_3}{2} = \frac{y_2 + y_4}{2} \]

4. **Eliminate the Fractions**: Multiply both sides of the equations by 2 to eliminate the fractions:
- From the x-coordinates:
\[ x_1 + x_3 = x_2 + x_4 \]
- From the y-coordinates:
\[ y_1 + y_3 = y_2 + y_4 \]

5. **Rearranging the Equations**: Rearranging both equations gives us:
- For x-coordinates:
\[ x_1 - x_2 + x_3 - x_4 = 0 \]
- For y-coordinates:
\[ y_1 - y_2 + y_3 - y_4 = 0 \]

6. **Conclusion**: The relationships derived from the coordinates of the vertices of the parallelogram can be summarized as:
- \( x_1 + x_3 = x_2 + x_4 \)
- \( y_1 + y_3 = y_2 + y_4 \)

Thus, the correct option that represents the relationship between the coordinates of the points in a parallelogram is:
\[ y_1 + y_3 = y_2 + y_4 \quad \text{and} \quad x_1 + x_3 = x_2 + x_4 \]