Three vertices of a parallelogram ABCD taken in order are A (3, 6), B (5, 10) and C (3, 2) find :
(i) the coordinates of the fourth vertex D.

Answer

To find the coordinates of the fourth vertex D of the parallelogram ABCD, we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Given Points:**
- A (3, 6)
- B (5, 10)
- C (3, 2)

2. **Let the Coordinates of D be (x, y).**

3. **Find the Midpoint O of AC:**
- The midpoint formula is given by:
\[ O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
- Here, A (3, 6) and C (3, 2) are the points.
- Substitute the coordinates into the formula:
\[ O = \left( \frac{3 + 3}{2}, \frac{6 + 2}{2} \right) = \left( \frac{6}{2}, \frac{8}{2} \right) = (3, 4) \]

4. **Set Up the Midpoint Equation for BD:**
- Since O is also the midpoint of BD, we can write:
\[ O = \left( \frac{5 + x}{2}, \frac{10 + y}{2} \right) \]
- We know O = (3, 4), so we can set up the equations:
\[ \frac{5 + x}{2} = 3 \quad \text{(1)} \]
\[ \frac{10 + y}{2} = 4 \quad \text{(2)} \]

5. **Solve Equation (1):**
- Multiply both sides by 2:
\[ 5 + x = 6 \]
- Subtract 5 from both sides:
\[ x = 1 \]

6. **Solve Equation (2):**
- Multiply both sides by 2:
\[ 10 + y = 8 \]
- Subtract 10 from both sides:
\[ y = -2 \]

7. **Conclusion:**
- The coordinates of the fourth vertex D are (1, -2).

### Final Answer:
The coordinates of the fourth vertex D are (1, -2).