Question

The points $(-1, 0)$ and $(-2, 1)$ are the two extremities of a diagonal of a parallelogram. If $(-6, 5)$ is the third vertex, then the fourth vertex of the parallelogram is:

• A. $(2, -6)$
• B. $(2, -5)$
• C. $(3, -4)$
• D. $(-3, 4)$
• E. $(3, -5)$

Hint:

Shortcut: \( D = A + C - B \). Just add diagonal endpoints and subtract the known vertex.

Answer 1

The Correct Option is D

Concept: In a parallelogram, diagonals bisect each other. So, midpoint of one diagonal = midpoint of the other diagonal.
• Midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
• Also: \( A + C = B + D \)

Step 1: Find midpoint of diagonal AC.
Given \( A(-1,0), \; C(-2,1) \) \[ M = \left( \frac{-1 + (-2)}{2}, \frac{0 + 1}{2} \right) = \left( \frac{-3}{2}, \frac{1}{2} \right) \]

Step 2: Use midpoint condition for BD.
Let \( B(-6,5) \), \( D(x,y) \) \[ \left( \frac{-6 + x}{2}, \frac{5 + y}{2} \right) = \left( \frac{-3}{2}, \frac{1}{2} \right) \]

Step 3: Solve for coordinates.
For \( x \): \[ -6 + x = -3 \Rightarrow x = 3 \] For \( y \): \[ 5 + y = 1 \Rightarrow y = -4 \]

Step 4: Final answer.
\[ \boxed{(3, -4)} \]