Show that the quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a rectangle is a rhombus.
Mid-point Theorem
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Show that the quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a rectangle is a rhombus.

KnowledgeBoat · Accepted Answer

Join BD. By mid-point theorem, The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it. In △ABC, Since, P and Q are the mid-points of AB and BC respectively. By mid-point theorem, ⇒ PQ || AC and PQ = AC ...(1) In △ADC, Since, S and R are the mid-points of AD and CD respectively. ⇒ SR || AC and SR = AC ...(2) From eq.(1) and (2), we have : PQ = SR and PQ || SR ...(3) In △ABD, Since, P and S are the mid-points of AB and AD respectively. PS || BD ⇒ PS = BD ...(4) In △BCD, Since, Q and R are the mid-points of BC and CD respectively. QR || BD ⇒ QR = BD ...(5) From eq.(4) and (5), we have: PS = QR and PS || QR ...(6) From eq.(3) and (6), we have: In quadrilateral PQRS opposite sides are parallel and equal. ∴ PQRS is a parallelogram. Given, ABCD is a rectangle. AB = CD and AD = BC In △ASP and △BQP, ⇒ AP = BP (P is the mid-point of AB) ⇒ AS = BQ (As, S and Q are mid-points of equal sides AD and BC respectively) ⇒ ∠SAP = ∠QBP (Both equal to 90°) ∴ △ASP ≅ △BQP (By S.A.S axiom) ⇒ PS = PQ ....(7) (Corresponding parts of congruent traingles are equal) From eq.(3), (6) and (7), we have: PS = PQ = QR = SR Since, all sides are equal and opposite sides are parallel, ∴ PQRS is a rhombus. Hence, the quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a rectangle is a rhombus.

Mathematics

Show that the quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a rectangle is a rhombus.

Mid-point Theorem

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