Question

The points $ (-a, -b), (0, 0), (a, b), \text{ and } (a^2, ab) $ are

• A. Vertical of a triangle
• B. Vertical of a square
• C. Vertical of a parallelogram
• D. Collinear

Hint:

To check if four points form a parallelogram, verify if the diagonals bisect each other.

Answer 1

The Correct Option is C

Step 1: Understanding the Geometry of the Points
The points are \( (-a, -b), (0, 0), (a, b), (a^2, ab) \).
To check if they form a parallelogram, we check if the diagonals bisect each other.
The midpoint of the diagonal joining \( (-a, -b) \) and \( (a, b) \) is: \[ \left( \frac{-a + a}{2}, \frac{-b + b}{2} \right) = (0, 0) \] The midpoint of the diagonal joining \( (0, 0) \) and \( (a^2, ab) \) is: \[ \left( \frac{0 + a^2}{2}, \frac{0 + ab}{2} \right) = \left( \frac{a^2}{2}, \frac{ab}{2} \right) \] These midpoints are the same, so the points form a parallelogram.
Step 2: Conclusion
Thus, the points form the vertical of a parallelogram.