If a̅, b̅, c̅, d̅ are the position vectors of the points A, B, C, D, respectively, such that no three of them are collinear and a̅ + c̅ = b̅ + d̅. then the quadrilateral ABCD is:

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This question was previously asked in

AAI ATC Junior Executive 21 Feb 2023 Shift 1 Official Paper

Answer 1

Given:

a̅, b̅, c̅, d̅ are the position vectors of the points A, B, C, D, respectively

no three of them are collinear and a̅ + c̅ = b̅ + d̅

Concept:

When the opposite sides of quadrilateral are equal and diagonals bisect each other then that is a parallelogram.

Calculation:

a̅ + c̅ = b̅ + d̅

⇒ c̅ - d̅ = b̅ - a̅

\(\rm \implies \vec{AB}=\vec{CD}\)

And a̅ + c̅ = b̅ + d̅

⇒ c̅ - b̅ = d̅ - a̅

\(\rm \implies \vec{AD}=\vec{BC}\)

Also, since a̅ + c̅ = b̅ + d̅

⇒ 1/2(a̅ + c̅ ) = 1/2(b̅ + d̅)

So, the position vector of mid point of BD = position vector of mid point of AC.

Hence the diagonal bisect each other.

then the given ABCD is a parallelogram.

Hence the option (4) is correct.

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