Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.
Mid-point Theorem
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Question

Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.

KnowledgeBoat · Accepted Answer

Let ABCD be the rectangle and P, Q, R and S be the mid-points of sides AB, BC, CD and DA respectively. Join PQRS. 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. Join AC and BD. We know that, Diagonals of a rectangle are equal. ∴ AC = BD = x (let) In Δ ABC, P and Q are the mid-points of sides AB and BC respectively. ∴ PQ || AC and PQ = (By mid-point theorem) ........(1) In Δ ADC, S and R are mid-points of sides AD and CD respectively. ∴ SR || AC and SR = (By mid-point theorem) .........(2) From equations (1) and (2), we get : PQ || SR and PQ = SR In Δ ABD, P and S are the mid-points of sides AB and AD respectively. ∴ PS || BD and PS = (By mid-point theorem) ........(3) In Δ BDC, Q and R are mid-points of sides BC and CD respectively. ∴ QR || BD and QR = (By mid-point theorem) .........(4) From equations (3) and (4), we get : PQ || SR and PS = QR By using equation (1), (2), (3) and (4), we get : ⇒ PQ = QR = SR = PS Since, opposite sides are parallel and all the sides are equal. ∴ PQRS is a rhombus. Hence, proved that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.

Mathematics

Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.

Mid-point Theorem

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Mathematics

Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.

Mid-point Theorem

Related Questions

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D, E and F are the mid-points of the sides AB, BC and CA of an isosceles △ ABC in which AB = BC. Prove that △ DEF is also isosceles.

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