Find the fourth vertex of the parallelogram whose consecutive vertices are (5,-1),(-3,-2),(9,12).

Answer

To find the fourth vertex of the parallelogram given the consecutive vertices \( A(5, -1) \), \( B(-3, -2) \), and \( C(9, 12) \), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-step Solution:

1. **Identify the Given Points**:
- Let \( A(5, -1) \), \( B(-3, -2) \), and \( C(9, 12) \).
- We need to find the fourth vertex \( D(x, y) \).

2. **Use the Midpoint Formula**:
- The midpoint \( M \) of diagonal \( AC \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
- For points \( A \) and \( C \):
\[ M = \left( \frac{5 + 9}{2}, \frac{-1 + 12}{2} \right) = \left( \frac{14}{2}, \frac{11}{2} \right) = (7, \frac{11}{2}) \]

3. **Set Up the Midpoint for Diagonal \( BD \)**:
- Since \( M \) is also the midpoint of diagonal \( BD \), we can use the midpoint formula again:
\[ M = \left( \frac{-3 + x}{2}, \frac{-2 + y}{2} \right) \]
- Setting this equal to the midpoint we found:
\[ \left( \frac{-3 + x}{2}, \frac{-2 + y}{2} \right) = (7, \frac{11}{2}) \]

4. **Create Equations**:
- From the x-coordinates:
\[ \frac{-3 + x}{2} = 7 \]
Multiplying both sides by 2:
\[ -3 + x = 14 \implies x = 14 + 3 = 17 \]
- From the y-coordinates:
\[ \frac{-2 + y}{2} = \frac{11}{2} \]
Multiplying both sides by 2:
\[ -2 + y = 11 \implies y = 11 + 2 = 13 \]

5. **Conclusion**:
- The coordinates of the fourth vertex \( D \) are \( (17, 13) \).

### Final Answer:
The fourth vertex \( D \) of the parallelogram is \( (17, 13) \).

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