Question

If A (-1, 2), B(2, - 1) and C (3, 1) are three vertices of a parallelogram, then the fourth vertex is

• A. D(-2, 0)
• B. D(0, 4)
• C. D(-2, 6)
• D. D(6, 2)

Answer 1

The Correct Option is B

In a parallelogram, the diagonals bisect each other. Therefore, the midpoint of diagonal AC should be the same as the midpoint of diagonal BD.
The midpoint of \( A(-1, 2) \) and \( C(3, 1) \) is: \[ \left( \frac{-1+3}{2}, \frac{2+1}{2} \right) = \left( 1, \frac{3}{2} \right) \] Now, let the coordinates of D be \( (x, y) \). The midpoint of \( B(2, -1) \) and \( D(x, y) \) is: \[ \left( \frac{2+x}{2}, \frac{-1+y}{2} \right) \] Equating the midpoints, we get: \[ \frac{2+x}{2} = 1 \quad \text{and} \quad \frac{-1+y}{2} = \frac{3}{2} \] Solving these equations: 1. \( \frac{2+x}{2} = 1 \implies x = 0 \) 2. \( \frac{-1+y}{2} = \frac{3}{2} \implies y = 4 \) 

The correct option is (B): \(D(0, 4)\)