Answer :

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






Magnitude PQ = AC


and





Magnitude SR = AC


Thus sides PQ and SR are equal and parallel


It shows PQRS is a parallelogram


Now,





By Dot product, we know




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



And



again by triangular law




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



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


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