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