The coordinates of three consecutive vertices of a parallelogram are (1, 3), (-1, 2) and (2, 5). Then find the coordinates of the fourth vertex.

Answer

To find the coordinates of the fourth vertex of a parallelogram given three consecutive vertices, we can use the properties of the midpoints of the diagonals. Let's denote the vertices as follows:

- A (1, 3)
- B (-1, 2)
- C (2, 5)
- D (x, y) - the fourth vertex we need to find.

### Step-by-step Solution:

1. **Identify the Given Points**:
We have the coordinates of three vertices:
- A = (1, 3)
- B = (-1, 2)
- C = (2, 5)

2. **Use the Midpoint Formula**:
The diagonals of a parallelogram bisect each other, meaning the midpoint of diagonal AC should equal the midpoint of diagonal BD.

The midpoint M of a segment with endpoints (x1, y1) and (x2, y2) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

3. **Calculate the Midpoint of AC**:
Using points A and C:
\[ M_{AC} = \left( \frac{1 + 2}{2}, \frac{3 + 5}{2} \right) = \left( \frac{3}{2}, \frac{8}{2} \right) = \left( \frac{3}{2}, 4 \right) \]

4. **Set Up the Midpoint of BD**:
The midpoint of BD (where B is (-1, 2) and D is (x, y)) is:
\[ M_{BD} = \left( \frac{-1 + x}{2}, \frac{2 + y}{2} \right) \]

5. **Set the Midpoints Equal**:
Since the midpoints are equal:
\[ \left( \frac{-1 + x}{2}, \frac{2 + y}{2} \right) = \left( \frac{3}{2}, 4 \right) \]

This gives us two equations:
- For the x-coordinates:
\[ \frac{-1 + x}{2} = \frac{3}{2} \]
- For the y-coordinates:
\[ \frac{2 + y}{2} = 4 \]

6. **Solve for x**:
From the equation \(\frac{-1 + x}{2} = \frac{3}{2}\):
\[ -1 + x = 3 \quad \Rightarrow \quad x = 4 \]

7. **Solve for y**:
From the equation \(\frac{2 + y}{2} = 4\):
\[ 2 + y = 8 \quad \Rightarrow \quad y = 6 \]

8. **Conclusion**:
The coordinates of the fourth vertex D are:
\[ D = (4, 6) \]

### Final Answer:
The coordinates of the fourth vertex D are (4, 6).