Find (x,y) if (3,2) (6,3) ,(x,y) and (6,5) are the vertices of a parallelogram :

Options

A. a. (5,6)
B. b. (6,5)
C. c. (9,6)
D. d. (9,5)

Answer

To find the coordinates (x, y) of the vertex of the parallelogram given the vertices (3, 2), (6, 3), and (6, 5), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the vertices:**
Let the vertices of the parallelogram be:
- A (3, 2)
- B (6, 3)
- C (x, y)
- D (6, 5)

2. **Find the midpoint of diagonal AC:**
The midpoint O of diagonal AC can be calculated using the midpoint formula:
\[ O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, A is (3, 2) and C is (x, y). Therefore, the coordinates of O are:
\[ O = \left( \frac{3 + x}{2}, \frac{2 + y}{2} \right) \]

3. **Find the midpoint of diagonal BD:**
Now, we calculate the midpoint of diagonal BD:
B is (6, 3) and D is (6, 5). The midpoint O of BD is:
\[ O = \left( \frac{6 + 6}{2}, \frac{3 + 5}{2} \right) = \left( 6, 4 \right) \]

4. **Set the midpoints equal to each other:**
Since the midpoints of the diagonals are the same, we can set the coordinates equal:
\[ \frac{3 + x}{2} = 6 \quad \text{(1)} \]
\[ \frac{2 + y}{2} = 4 \quad \text{(2)} \]

5. **Solve for x using equation (1):**
Multiply both sides of equation (1) by 2:
\[ 3 + x = 12 \]
Subtract 3 from both sides:
\[ x = 12 - 3 = 9 \]

6. **Solve for y using equation (2):**
Multiply both sides of equation (2) by 2:
\[ 2 + y = 8 \]
Subtract 2 from both sides:
\[ y = 8 - 2 = 6 \]

7. **Conclusion:**
Therefore, the coordinates (x, y) are:
\[ (x, y) = (9, 6) \]

### Final Answer:
The coordinates (x, y) are (9, 6).