Question

Let ABCD be a parallelogram such that the coordinates of its three vertices A, B, C are (1, 1), (3, 4), and (−2, 8), respectively. Then, the coordinates of the vertex D are

• A. (−4, 5)
• B. (4, 5)
• C. (−3, 4)
• D. (0, 11)

Answer 1

The Correct Option is A

Given

Parallelogram ABCD with vertices:

• \( A = (1, 1) \)
• \( B = (3, 4) \)
• \( C = (-2, 8) \)
• \( D = (x, y) \) — to be found

In a parallelogram, diagonals bisect each other.

Step 1: Use Midpoint of Diagonal AC

Find the midpoint of diagonal AC: \[ \text{Midpoint}_{AC} = \left( \frac{1 + (-2)}{2}, \frac{1 + 8}{2} \right) = \left( \frac{-1}{2}, \frac{9}{2} \right) \]

 Step 2: Let Point D Be \( (x, y) \)

Midpoint of diagonal BD is: \[ \text{Midpoint}_{BD} = \left( \frac{3 + x}{2}, \frac{4 + y}{2} \right) \] Since diagonals bisect each other: \[ \frac{3 + x}{2} = \frac{-1}{2} \quad \text{and} \quad \frac{4 + y}{2} = \frac{9}{2} \]

Step 3: Solve the Equations

First equation: \[ \frac{3 + x}{2} = \frac{-1}{2} \Rightarrow 3 + x = -1 \Rightarrow x = -4 \] 
Second equation: \[ \frac{4 + y}{2} = \frac{9}{2} \Rightarrow 4 + y = 9 \Rightarrow y = 5 \]

 Final Answer

Therefore, the coordinates of point \( D \) are: \[ \boxed{(-4, 5)} \]

Answer 2

 Given:

• Parallelogram ABCD
• Vertices: \( A = (1, 1),\ B = (3, 4),\ C = (-2, 8) \)
• Find coordinates of vertex \( D = (x, y) \)

Diagonals Bisect Each Other

In a parallelogram, the diagonals bisect each other. 
Therefore, the midpoint of diagonal AC must equal the midpoint of diagonal BD.

 Step 1: Midpoint of AC

\[ \text{Midpoint}_{AC} = \left( \frac{1 + (-2)}{2}, \frac{1 + 8}{2} \right) = \left( \frac{-1}{2}, \frac{9}{2} \right) \]

 Step 2: Midpoint of BD (Let D = (x, y))

\[ \text{Midpoint}_{BD} = \left( \frac{3 + x}{2}, \frac{4 + y}{2} \right) \] 
Equating midpoints: \[ \frac{3 + x}{2} = \frac{-1}{2}, \quad \frac{4 + y}{2} = \frac{9}{2} \]

Step 3: Solving the Equations

Solve the first equation: \[ 3 + x = -1 \Rightarrow x = -4 \] Solve the second equation: \[ 4 + y = 9 \Rightarrow y = 5 \]

 Final Answer

Therefore, the coordinates of point \( D \) are: \[ \boxed{(-4, 5)} \]