The diagonals of a quadrilateral ABCD are perpendicular to each other. Prove that the quadrilateral obtained by joining the mid-points of its adjacent sides is a rectangle.
Mid-point Theorem
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Question

The diagonals of a quadrilateral ABCD are perpendicular to each other. Prove that the quadrilateral obtained by joining the mid-points of its adjacent sides 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. Let ABCD be a quadrilateral where P, Q, R and S are the mid-point of AB, BC, CD and DA. In △ ABC, P and Q are mid-points of AB and BC respectively. ⇒ PQ = and PQ || AC. [By mid-point theorem] .......(1) In △ ADC, S and R are mid-points of AD and CD respectively. ⇒ SR = and SR || AC. [By mid-point theorem] .......(2) From (1) and (2), we get : PQ = SR and PQ || SR. In △ BCD, R and Q are mid-points of CD and BC respectively. ⇒ QR = and QR || BD. [By mid-point theorem] .......(3) In △ ABD, S and P are mid-points of AD and AB respectively. ⇒ PS = and PS || BD. [By mid-point theorem] .......(4) From (3) and (4), we get : QR = PS and QR || PS. Since, diagonals of quadrilateral intersect at right angle. ∴ ∠AOD = ∠COD = AOB = ∠BOC = 90°. From figure, PQ || AC ∴ ∠PXO = ∠AOD = 90° (Corresponding angles are equal) ∴ ∠QXO = ∠COD = 90° (Corresponding angles are equal) SR || AC ∴ ∠SZO = ∠AOB = 90° (Corresponding angles are equal) ∴ ∠RZO = ∠BOC = 90° (Corresponding angles are equal) PS || BD ∴ ∠S = ∠RZO = 90° (Corresponding angles are equal) ∴ ∠P = ∠QXO = 90° (Corresponding angles are equal) QR || BD ∴ ∠R = ∠SZO = 90° (Corresponding angles are equal) ∴ ∠Q = ∠PXO = 90° (Corresponding angles are equal) Since, in quadrilateral PQRS, Each interior angle is equal to 90° and opposite sides are parallel and equal. ∴ PQRS is a rectangle. Hence, proved that the the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.

Mathematics

The diagonals of a quadrilateral ABCD are perpendicular to each other. Prove that the quadrilateral obtained by joining the mid-points of its adjacent sides is a rectangle.

Mid-point Theorem

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