A parallelogram has three consecutive vertices (-3, 4), (0, -4) and (5,2). The fourth vertex is

Options

A. (2, 10)
B. (2, 9)
C. (3, 9)
D. (4, 10)

Answer

To find the fourth vertex of the parallelogram given three consecutive vertices, we can follow these steps:

### Step 1: Identify the given vertices
Let the three consecutive vertices of the parallelogram be:
- \( A(-3, 4) \)
- \( B(0, -4) \)
- \( C(5, 2) \)

### Step 2: Use the midpoint formula
In a parallelogram, the diagonals bisect each other. Therefore, the midpoint of diagonal \( AC \) should equal the midpoint of diagonal \( BD \) (where \( D \) is the fourth vertex we need to find).

The midpoint \( O \) of diagonal \( AC \) can be calculated using the midpoint formula:
\[ O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( A(x_1, y_1) \) and \( C(x_2, y_2) \).

Substituting the coordinates of \( A \) and \( C \):
\[ O = \left( \frac{-3 + 5}{2}, \frac{4 + 2}{2} \right) = \left( \frac{2}{2}, \frac{6}{2} \right) = (1, 3) \]

### Step 3: Set up the equation for the midpoint of diagonal \( BD \)
Let \( D \) have coordinates \( (x, y) \). The midpoint \( O \) of diagonal \( BD \) is given by:
\[ O = \left( \frac{0 + x}{2}, \frac{-4 + y}{2} \right) \]
Setting this equal to the midpoint we found earlier:
\[ \left( \frac{0 + x}{2}, \frac{-4 + y}{2} \right) = (1, 3) \]

### Step 4: Solve for \( x \) and \( y \)
From the first coordinate:
\[ \frac{x}{2} = 1 \implies x = 2 \]

From the second coordinate:
\[ \frac{-4 + y}{2} = 3 \implies -4 + y = 6 \implies y = 10 \]

### Step 5: Conclusion
Thus, the coordinates of the fourth vertex \( D \) are:
\[ D(2, 10) \]

### Final Answer
The fourth vertex of the parallelogram is \( (2, 10) \).

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