If A (3, 5), B (-3, - 4), C (7, 10) are the vertices of a parallelogram taken in the order, then the co-ordinates of the fourth vertex are

Options

A. (13, 19)
B. (15, 10)
C. (19, 10)
D. (15, 19)

Answer

To find the coordinates of the fourth vertex \( D(x, y) \) of the parallelogram formed by the vertices \( A(3, 5) \), \( B(-3, -4) \), and \( C(7, 10) \), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Coordinates of the Given Points:**
- Let \( A(3, 5) \)
- Let \( B(-3, -4) \)
- Let \( C(7, 10) \)
- Let \( D(x, y) \) be the unknown vertex.

2. **Use the Midpoint Formula:**
The midpoint \( M \) of diagonal \( AC \) should be equal to the midpoint of diagonal \( BD \).
- The midpoint \( M_{AC} \) of \( A \) and \( C \) is given by:
\[ M_{AC} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{3 + 7}{2}, \frac{5 + 10}{2} \right) = \left( \frac{10}{2}, \frac{15}{2} \right) = (5, 7.5) \]

3. **Set Up the Midpoint for \( BD \):**
The midpoint \( M_{BD} \) of \( B \) and \( D \) is given by:
\[ M_{BD} = \left( \frac{-3 + x}{2}, \frac{-4 + y}{2} \right) \]
Since \( M_{AC} = M_{BD} \), we can set the coordinates equal:
\[ \frac{-3 + x}{2} = 5 \quad \text{(1)} \]
\[ \frac{-4 + y}{2} = 7.5 \quad \text{(2)} \]

4. **Solve for \( x \) from Equation (1):**
\[ -3 + x = 10 \quad \Rightarrow \quad x = 10 + 3 = 13 \]

5. **Solve for \( y \) from Equation (2):**
\[ -4 + y = 15 \quad \Rightarrow \quad y = 15 + 4 = 19 \]

6. **Conclusion:**
The coordinates of the fourth vertex \( D \) are \( (13, 19) \).

### Final Answer:
The coordinates of the fourth vertex \( D \) are \( (13, 19) \).

---