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

Updated On: Jul 26, 2025
  • (−4, 5)

  • (4, 5) 

  • (−3, 4)

  • (0, 11) 

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The Correct Option is A

Approach Solution - 1

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: \[ \boxed{(-4, 5)} \]

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Approach Solution -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: \[ \boxed{(-4, 5)} \]

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