What is the fourth vertex of a parallelogram, if its three consecutive vertices are (-2,-1), (1,0) and (4,3)?

Options

A. `(0,-2)`
B. `(2,2)`
C. `(2,1)`
D. `(1,2)`

Answer

To find the fourth vertex of a parallelogram given three consecutive vertices, we can use the property that the diagonals of a parallelogram bisect each other. Let's denote the given vertices as follows:

- A = (-2, -1)
- B = (1, 0)
- C = (4, 3)

We need to find the fourth vertex D, which we will denote as (x, y).

### Step 1: Find the midpoint of diagonal AC
The midpoint M of diagonal AC can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( (x_1, y_1) \) are the coordinates of point A and \( (x_2, y_2) \) are the coordinates of point C.

Substituting the coordinates:
\[ M = \left( \frac{-2 + 4}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{2}{2}, \frac{2}{2} \right) = (1, 1) \]

### Step 2: Set up the midpoint equation for diagonal BD
Since the diagonals bisect each other, the midpoint of diagonal BD must also be M. Using the coordinates of point B and point D, we can set up the following equations:
\[ M = \left( \frac{1 + x}{2}, \frac{0 + y}{2} \right) \]
This means:
\[ \frac{1 + x}{2} = 1 \quad \text{(1)} \]
\[ \frac{0 + y}{2} = 1 \quad \text{(2)} \]

### Step 3: Solve for x using equation (1)
From equation (1):
\[ \frac{1 + x}{2} = 1 \]
Multiplying both sides by 2:
\[ 1 + x = 2 \]
Subtracting 1 from both sides:
\[ x = 1 \]

### Step 4: Solve for y using equation (2)
From equation (2):
\[ \frac{0 + y}{2} = 1 \]
Multiplying both sides by 2:
\[ y = 2 \]

### Step 5: Write the coordinates of the fourth vertex D
Now that we have both x and y, the coordinates of the fourth vertex D are:
\[ D = (1, 2) \]

### Conclusion
Thus, the fourth vertex of the parallelogram is \( D(1, 2) \).

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