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


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


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


Magnitude SR = AC

Thus sides PQ and SR are equal and parallel

It shows PQRS is a parallelogram


By Dot product, we know

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


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

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses

Write a unit vectMathematics - Board Papers

Fill in the blankMathematics - Exemplar

A and B areMathematics - Board Papers

L and M are two pMathematics - Board Papers

The scalar producMathematics - Board Papers

A vector <span laMathematics - Exemplar