If three vertices of a parallelogram are A(3, 5) B(-5, -4) and C(7, 10), then fourth vertex is

Options

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

Answer

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

Let's denote the fourth vertex as D(x, y).

### Step-by-Step Solution:

1. **Identify the Midpoints of the Diagonals**:
The midpoints of the diagonals AC and BD must be equal. We will first find the midpoint of AC and then set it equal to the midpoint of BD.

2. **Calculate the Midpoint of AC**:
The formula for the midpoint M of two points (x1, y1) and (x2, y2) is:
\[ M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right) \]
For points A(3, 5) and C(7, 10):
\[ M_{AC} = \left(\frac{3 + 7}{2}, \frac{5 + 10}{2}\right) = \left(\frac{10}{2}, \frac{15}{2}\right) = (5, 7.5) \]

3. **Set Up the Midpoint of BD**:
The midpoint of BD is given by:
\[ M_{BD} = \left(\frac{-5 + x}{2}, \frac{-4 + y}{2}\right) \]
Since M_{AC} = M_{BD}, we can set the midpoints equal to each other:
\[ \left(5, 7.5\right) = \left(\frac{-5 + x}{2}, \frac{-4 + y}{2}\right) \]

4. **Create Equations from the Midpoints**:
From the equality of midpoints, we can create two equations:
- For the x-coordinates:
\[ 5 = \frac{-5 + x}{2} \]
- For the y-coordinates:
\[ 7.5 = \frac{-4 + y}{2} \]

5. **Solve for x**:
Multiply both sides of the first equation by 2:
\[ 10 = -5 + x \]
Adding 5 to both sides gives:
\[ x = 15 \]

6. **Solve for y**:
Multiply both sides of the second equation by 2:
\[ 15 = -4 + y \]
Adding 4 to both sides gives:
\[ y = 19 \]

7. **Conclusion**:
The coordinates of the fourth vertex D are (15, 19).

### Final Answer:
The fourth vertex D is at the coordinates (15, 19).

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