Question:medium

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: Jan 15, 2026

  • (−4, 5)

  • (4, 5) 

  • (−3, 4)

  • (0, 11) 

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The Correct Option is A

Solution and Explanation

Given

Parallelogram ABCD with vertices:

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

The diagonals of a parallelogram bisect each other.

Step 1: Midpoint of Diagonal AC

Calculate 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: Coordinates of Point D

The midpoint of diagonal BD is: \[ \text{Midpoint}_{BD} = \left( \frac{3 + x}{2}, \frac{4 + y}{2} \right) \] As the diagonals bisect each other, their midpoints are identical: \[ \frac{3 + x}{2} = \frac{-1}{2} \quad \text{and} \quad \frac{4 + y}{2} = \frac{9}{2} \]

Step 3: Solve for x and y

Solve the equation for x: \[ \frac{3 + x}{2} = \frac{-1}{2} \Rightarrow 3 + x = -1 \Rightarrow x = -4 \]

Solve the equation for y: \[ \frac{4 + y}{2} = \frac{9}{2} \Rightarrow 4 + y = 9 \Rightarrow y = 5 \]

 Final Answer

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

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In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
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