P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle.
Under what condition will PQRS be a square ?

Question

P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle. Under what condition will PQRS be a square ?

KnowledgeBoat · Accepted Answer

Given: In rhombus ABCD, P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively. To prove: PQRS is a rectangle. Construction: Join PQ, QR, RS, and SP to form quadrilateral PQRS. Also, join diagonals AC and BD of rhombus ABCD. Proof: Since ABCD is a rhombus, its diagonals AC and BD, bisect each other at right angles. ⇒ ∠AOD = ∠COD = ∠BOC = ∠AOB = 90° In Δ ACD, S and R are the midpoints of AD and CD. By the midpoint theorem, SR ∥ AC and SR = AC ...................(1) Similarly, in Δ ABC, P and Q are the midpoints of AB and BC. By the midpoint theorem, PQ ∥ AC and PQ = AC ...................(2) From equations (1) and (2), SR ∥ PQ and SR = PQ = AC ...................(3) In the same way, in Δ ABD, S and P are the midpoints of AD and AB. By the midpoint theorem, SP ∥ BD and SP = BD ...................(4) Similarly, in Δ BDC, R and Q are the midpoints of DC and BC. By the midpoint theorem, RQ ∥ BD and RQ = BD ...................(5) From equations (4) and (5), SP ∥ RQ and SP = RQ = BD ...............(6) Since opposite sides are equal and parallel, PQRS is a parallelogram. Also, since diagonals of rhombus bisect each other at right angles, the midpoints P, Q, R, S form a quadrilateral where: ∠SPQ = ∠PQR = ∠QRS = ∠RSP = 90° Thus, PQRS is a rectangle. For PQRS to be a square, it must be a rectangle with all sides equal: PQ = QR = RS = SP From the Midpoint Theorem, we know: PQ = AC, QR = BD For PQRS to be a square, we must have: AC = BD ⇒ AC = BD Since AC and BD are diagonals of rhombus ABCD, AC = BD is only true when ABCD is a square. Thus, PQRS is a square if and only if ABCD is a square. Hence, PQRS is a rectangle and PQRS will be a square when ABCD is a square.

Mathematics

P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle.
Under what condition will PQRS be a square ?

Mid-point Theorem

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Mathematics

P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle.Under what condition will PQRS be a square ?

Mid-point Theorem

Related Questions

In △ ABC, D, E and F are mid-points of sides AB, BC and AC respectively. Prove that AE and DF bisect each other.

In a kite shaped figure ABCD, AB = AD and CB = CD. Points P, Q and R are mid-points of sides AB, BC and CD respectively.Prove that :(i) ∠PQR = 90°(ii) Line through P and parallel to QR bisects side AD.

In a scalene triangle ABC, AD and BE are medians. F is a point on AC so that DF//BE. Show that : AC = 4 x EF.

Show that the quadrilateral formed by joining the mid-points of the adjacent sides of a square, is also a square.

P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle.
Under what condition will PQRS be a square ?

In a kite shaped figure ABCD, AB = AD and CB = CD. Points P, Q and R are mid-points of sides AB, BC and CD respectively.
Prove that :
(i) ∠PQR = 90°
(ii) Line through P and parallel to QR bisects side AD.