If A (1,0), B (5,3), C(2,7) and D (x, y) are vertices of a parallelogram ABCD, the co-ordinates of D are

Options

A. (-2, -3)
B. (-2, 4)
C. (2, -3)
D. (3, 5)

Answer

To find the coordinates of point D (x, y) in the parallelogram ABCD with given vertices A(1, 0), B(5, 3), and C(2, 7), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Midpoint of Diagonal AC**:
The midpoint O of diagonal AC can be calculated using the midpoint formula:
\[ O = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Here, A(1, 0) and C(2, 7) are the points.
\[ O = \left(\frac{1 + 2}{2}, \frac{0 + 7}{2}\right) = \left(\frac{3}{2}, \frac{7}{2}\right) \]

2. **Identify the Midpoint of Diagonal BD**:
Similarly, the midpoint O of diagonal BD can be calculated as:
\[ O = \left(\frac{5 + x}{2}, \frac{3 + y}{2}\right) \]
Here, B(5, 3) and D(x, y) are the points.

3. **Set the Midpoints Equal**:
Since both midpoints are the same (O), we can set the x-coordinates and y-coordinates equal to each other:
\[ \frac{3}{2} = \frac{5 + x}{2} \]
\[ \frac{7}{2} = \frac{3 + y}{2} \]

4. **Solve for x**:
From the first equation:
\[ 3 = 5 + x \implies x = 3 - 5 = -2 \]

5. **Solve for y**:
From the second equation:
\[ 7 = 3 + y \implies y = 7 - 3 = 4 \]

6. **Conclusion**:
Therefore, the coordinates of point D are:
\[ D(-2, 4) \]

### Final Answer:
The coordinates of D are (-2, 4).

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