Question

Let A, B, C, D be the points in the plane with position vectors \(-2\hat{i}-\hat{j}\), \(4\hat{i}\), \(3\hat{i}+3\hat{j}\) and \(-3\hat{i}+2\hat{j}\) respectively, then \(◻\)ABCD is

• A. a parallelogram which is neither a rhombus nor a reactangle
• B. a square
• C. a rectangle but not a square
• D. a rhombus but not a square

Hint:

Compare opposite sides, then adjacent side lengths and angle.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Points: A(-2, -1), B(4, 0), C(3, 3), D(-3, 2). Check vectors of opposite sides to test for a parallelogram.

Step 2: Opposite sides:
\(\overrightarrow{AB} = (6, 1)\), \(\overrightarrow{DC} = (6, 1)\). So \(AB\) and \(DC\) are equal and parallel, hence ABCD is a parallelogram.

Step 3: Adjacent sides:
\(\overrightarrow{AD} = (-1, 3)\). \(|AB| = \sqrt{37}\), \(|AD| = \sqrt{10}\), which are different, so it is not a rhombus.
\(\overrightarrow{AB}\cdot\overrightarrow{AD} = -6 + 3 = -3 \neq 0\), so the angle is not a right angle and it is not a rectangle.

Step 4: Result:
ABCD is a parallelogram that is neither a rhombus nor a rectangle, option (A). Since it is not a rectangle or rhombus, it cannot be a square either.

Final Answer:
Opposite sides match, but sides and angle are unequal. \[ \boxed{\text{(A) }\text{Parallelogram, not rhombus or rectangle}} \]