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

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

KnowledgeBoat · Accepted Answer

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. PQ || AC ⇒ PQ = AC ...(1) In △ADC, Since, S and R are the mid-points of AD and CD respectively. SR || AC ⇒ SR = AC ...(2) From eq.(1) and (2), we have: 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 ...(6) From eq.(3) and (6), we have: ∴ PQRS is a parallelogram. We know that, Diagonals of rhombus intersect at right angles. ⇒ ∠EOF = 90° In quadrilateral OERF, ER || OF and EO || RF ∴ OERF is a parallelogram. Opposite angles of a parallelogram are equal. ⇒ ∠EOF = ∠ERF = 90° In parallelogram PQRS, ⇒ ∠QRS = ∠QPS = 90° ⇒ ∠PSR = ∠PQR = x (let) ∠QRS + ∠QPS + ∠PSR + ∠PQR = 360° ⇒ 90° + 90° + x + x = 360° ⇒ 180° + 2x = 360° ⇒ 2x = 360° - 180° ⇒ 2x = 180° ⇒ x = ⇒ x = 90° ⇒ ∠PSR = ∠PQR = 90° Since, in parallelogram PQRS, opposite sides are equal and parallel and all the interior angles equal to 90°. ∴ PQRS is a rectangle. Hence, proved that the quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a rhombus is a rectangle.

Mathematics

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

Mid-point Theorem

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