Question

ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Prove that the quadrilateral PQRS is a rectangle. 

Answer 1

According to the given statement, the figure will be a shown alongside; using mid-point theorem:- 

In ∆ABC, PQ ║ AC and PQ = 1/2 AC ....(i) 

 In ∆ADC, SR ║ AC and SR = 1/2 AC ....(ii) 

∴ P = SR and PQ ║ SR [From (i) and (ii)] 

⇒ PQRS is a parallelogram. 

Now, PQRS will be a rectangle if any angle of the parallelogram PWRS is 90⁰ 

PQ ║ AC [By mid-point theorem] 

QR = BD [By mid-point theorem] 

But, AC ⊥ BD [Diagonals of a rhombus are perpendicular to each other] 

∴ PQ ⊥ QR [Angle between two lines = angles between their parallels] 

⇒ PQRS is a rectangle