Question

The fourth vertex D of a parallelogram ABCD whose three vertices are A\(-4, 1\), B\(4, 5\) and C\(6, 1\) is :

• A. \(-2, -3\)
• B. \(3, -2\)
• C. \(0, -1\)
• D. \(0, 1\)

Hint:

For any parallelogram \(ABCD\) in order, the sum of coordinates of opposite vertices is equal because of the midpoint property:
\[ x_A + x_C = x_B + x_D \implies x_D = x_A + x_C - x_B \]
\[ y_D = y_A + y_C - y_B \]
Let's check this shortcut:
\[ x_D = -4 + 6 - 4 = -2 \]
\[ y_D = 1 + 1 - 5 = -3 \]
This gives the coordinates \((-2, -3)\) in seconds without needing midpoint formulas.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given three vertices of a parallelogram \(ABCD\), namely \(A(-4, 1)\), \(B(4, 5)\), and \(C(6, 1)\). We need to determine the coordinates of the fourth vertex \(D\).

Step 2: Key Formula or Approach:
A fundamental property of a parallelogram is that its diagonals bisect each other. Therefore, the midpoint of diagonal \(AC\) must be equal to the midpoint of diagonal \(BD\).
The formula for the midpoint of a line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Step 3: Detailed Explanation:
1. Let the coordinates of the fourth vertex \(D\) be \((x, y)\).
2. Find the coordinates of the midpoint of diagonal \(AC\) joining \(A(-4, 1)\) and \(C(6, 1)\):
\[ \text{Midpoint of } AC = \left( \frac{-4 + 6}{2}, \frac{1 + 1}{2} \right) = \left( \frac{2}{2}, \frac{2}{2} \right) = (1, 1) \]
3. Find the coordinates of the midpoint of diagonal \(BD\) joining \(B(4, 5)\) and \(D(x, y)\):
\[ \text{Midpoint of } BD = \left( \frac{4 + x}{2}, \frac{5 + y}{2} \right) \]
4. Since the midpoint of \(AC\) is the same as the midpoint of \(BD\), we equate their coordinates:
Equating the \(x\)-coordinates:
\[ \frac{4 + x}{2} = 1 \]
\[ 4 + x = 2 \implies x = 2 - 4 = -2 \]
Equating the \(y\)-coordinates:
\[ \frac{5 + y}{2} = 1 \]
\[ 5 + y = 2 \implies y = 2 - 5 = -3 \]
5. Combining these values gives the coordinates of vertex \(D\) as \((-2, -3)\).

Step 4: Final Answer:
The coordinates of the fourth vertex \(D\) are \((-2, -3)\), which corresponds to option (A).