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

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 square in which E, F, G and H are mid-points of AB, BC, CD and DA respectively. We know that diagonals of a square are equal and bisect each other. AC = BD AO = OC = BO = OD Join EF, FG, GH and HE. Join AC and BD. In △ACD, G and H are mid-points of CD and AD respectively. By mid-point theorem, ∴ GH || AC and GH = AC GH = AO (∵ O is the mid-point of AC) ...(1) In △ABC, E and F are mid-points of AB and BC respectively. ∴ EF || AC and EF = AC EF = AO (∵ O is the mid-point of AC) ...(2) In △ABD, E and H are mid-points of AB and AD respectively. ∴ EH || BD and EH = BD EH = BO (∵ O is the mid-point of BD) ∴ EH = AO ...(3) In △BCD, G and F are mid-points of CD and BC respectively. ∴ FG || BD and FG = BD FG = BO (∵ O is the mid-point of BD) ∴ FG = AO ....(4) From eq.(1), (2), (3) and (4), we get: EH || FG, EF || GH and EH = FG = GH = EF Since, both the opposite sides of a quadrilateral are parallel. ∴ EFGH is a parallelogram. In △GOH and △GOF, ⇒ OH = OF (Diagonals of parallelogram bisect each other) ⇒ OG = OG (Common side) ⇒ GH = GF (Proved above) ∴ △GOH ≅ △GOF (S.S.S axiom) ⇒ ∠GOH = ∠GOF (Corresponding parts of congruent triangles are equal) From figure, ⇒ ∠GOH + ∠GOF = 180° ⇒ ∠GOH + ∠GOH = 180° ⇒ 2∠GOH = 180° ⇒ ∠GOH = ⇒ ∠GOH = 90° So, the diagonals of EFGH bisect and are perpendicular to each other and all sides of quadrilateral EFGH are equal. ∴ EFGH is a square. Hence, the quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a square is a square.

Mathematics

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

Mid-point Theorem

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