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

Options

A. (5, 6)
B. (6, 5)
C. (9, 6)
D. (9, 5)

Answer

To find the coordinates (x, y) of the vertex of the rectangle given the vertices (3, 2), (6, 3), (x, y), and (6, 5), we can follow these steps:

### Step 1: Identify the given vertices
The vertices of the rectangle are:
- A (3, 2)
- B (6, 3)
- C (x, y)
- D (6, 5)

### Step 2: Find the midpoint of the diagonal
In a rectangle, the diagonals bisect each other. We can find the midpoint M of the diagonal formed by points B (6, 3) and D (6, 5).

The formula for the midpoint M of two points (x1, y1) and (x2, y2) is:
\[ M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right) \]

Substituting the coordinates of B and D:
\[ M = \left(\frac{6 + 6}{2}, \frac{3 + 5}{2}\right) = \left(\frac{12}{2}, \frac{8}{2}\right) = (6, 4) \]

### Step 3: Use the midpoint to find (x, y)
Since M is also the midpoint of the diagonal AC, we can set up the equation for the midpoint of A (3, 2) and C (x, y):
\[ M = \left(\frac{3 + x}{2}, \frac{2 + y}{2}\right) \]

We already found that M = (6, 4). Therefore, we can set up the following equations:
1. \(\frac{3 + x}{2} = 6\)
2. \(\frac{2 + y}{2} = 4\)

### Step 4: Solve for x
From the first equation:
\[ \frac{3 + x}{2} = 6 \]
Multiplying both sides by 2:
\[ 3 + x = 12 \]
Subtracting 3 from both sides:
\[ x = 12 - 3 = 9 \]

### Step 5: Solve for y
From the second equation:
\[ \frac{2 + y}{2} = 4 \]
Multiplying both sides by 2:
\[ 2 + y = 8 \]
Subtracting 2 from both sides:
\[ y = 8 - 2 = 6 \]

### Final Answer
Thus, the coordinates (x, y) are:
\[ (x, y) = (9, 6) \]

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