The extremities of the diagonal of a parallelogram are the points (3,-4) and (-6,5). Third vertex is the point (-2,1), then the fourth vertex is

Options

A. (1,1)
B. (1,0)
C. (0,1)
D. (-1,0)

Answer

To find the fourth vertex of the parallelogram given the extremities of the diagonal and one vertex, we can follow these steps:

### Step 1: Identify the given points
The extremities of the diagonal are:
- Point A (3, -4)
- Point C (-6, 5)

The third vertex (Point B) is given as:
- Point B (-2, 1)

### Step 2: Use the property of the diagonals of a parallelogram
In a parallelogram, the diagonals bisect each other. Therefore, the midpoint of diagonal AC should be equal to the midpoint of diagonal BD.

### Step 3: Calculate the midpoint of diagonal AC
The midpoint M of segment AC can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points A and C:
\[ M = \left( \frac{3 + (-6)}{2}, \frac{-4 + 5}{2} \right) = \left( \frac{-3}{2}, \frac{1}{2} \right) \]

### Step 4: Set up the equation for midpoint of diagonal BD
Let the coordinates of the fourth vertex D be (x, y). The midpoint of diagonal BD is given by:
\[ M = \left( \frac{-2 + x}{2}, \frac{1 + y}{2} \right) \]

### Step 5: Equate the midpoints
Since the midpoints of both diagonals are equal, we can set the midpoints equal to each other:
\[ \frac{-2 + x}{2} = \frac{-3}{2} \]
\[ \frac{1 + y}{2} = \frac{1}{2} \]

### Step 6: Solve for x and y
1. From the first equation:
\[ -2 + x = -3 \implies x = -3 + 2 \implies x = -1 \]

2. From the second equation:
\[ 1 + y = 1 \implies y = 1 - 1 \implies y = 0 \]

### Step 7: Write the coordinates of the fourth vertex
Thus, the coordinates of the fourth vertex D are:
\[ D = (-1, 0) \]

### Final Answer
The fourth vertex of the parallelogram is (-1, 0).

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