Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.

Given:- ABCD is a rectangle

To prove:- PQRS is rhombus thus finding its properties in PQRS

i.e. All sides equal and parallel

Let, P, Q, R and S are midpoints of sides AB, BC, CD and DA respectively

Therefore

also AB = CD, BC = AD (ABCD is rectangle opposite sides are equal)

Therefore

AP = PB = DR = RC and BQ = QC = AS = SD ……(i)

IMP:- Direction/arrow head of vector should be placed correctly

Now, considering in vector notion and applying triangular law of vector addition, we get

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Magnitude PQ = AC

and

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Magnitude SR = AC

Thus sides PQ and SR are equal and parallel

It shows PQRS is a parallelogram

Now,

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By Dot product, we know

Here ABCD is rectangle and have 90° at A, B, C, D

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And

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again by triangular law

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⇒

By Dot product, we know

Here ABCD is rectangle and have 90° at A, B, C, D

⇒

From above similarities of sides of rectangle in eq (i), we have

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Hence PQ = PS

And from above results we have

All sides of parallelogram are equal

PQ = QR = RS = SP

Hence proved by property of rhombus (all sides are equal and opposite sides are parallel), PQRS is rhombus