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

Let ABCD be a square in which E, F, G and H are midpoints of AB, BC, CD and DA respectively. Join EF, FG, GH and HE. Join AC and BD. In △ACD, G and H are mid-points of CD and AD respectively, ∴ GH || AC and GH = AC .......(i) In △ABC, E and F are mid-points of AB and BC respectively, ∴ EF || AC and EF = AC .......(ii) Using (i) and (ii) we get, EF || GH and EF = GH = AC ........(1) In △ABD, E and H are mid-points of AB and AD respectively, ∴ EH || BD and EH = BD .......(iii) In △BCD, G and F are mid-points of CD and BC respectively, ∴ FG || BD and FG = BD .......(iv) Using (iii) and (iv) we get, EH || FG and EH = FG = BD ........(2) We know that diagonals of square are equal, AC = BD Dividing both sides by 2 we get, Substituting above value in 1 and 2 we get, EF = GH = EH = FG .........(v) ∴ EFGH is a parallelogram. In △GOH and △GOF, OH = OF as diagonals of parallelogram bisect each other. OG = OG (Common) GH = GF (From (v)) ∴ △GOH ≅ △GOF (SSS axiom of congruency) ∠GOH = ∠GOF (c.p.c.t.c.) From figure, ⇒ ∠GOH + ∠GOF = 180° ⇒ ∠GOH + ∠GOH = 180° ⇒ 2∠GOH = 180° ⇒ ∠GOH = 90°. So, the diagonals of EFGH bisect and are perpendicular to each other. ∴ EFGH is a square. Hence, proved that quadrilateral formed by joining the mid-points of the adjacent sides of a square, is also a square.

In the adjoining figure, ABCD is a parallelogram. E and F are mid-points of the sides AB and CD respectively. The straight lines AF and BF meet the straight lines ED and EC in points G and H respectively. Prove that
(i) △HEB ≅ △HCF
(ii) GEHF is a parallelogram.

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Mathematics

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

Mid-point Theorem

Related Questions

In the adjoining figure, ABCD is a parallelogram. E and F are mid-points of the sides AB and CD respectively. The straight lines AF and BF meet the straight lines ED and EC in points G and H respectively. Prove that
(i) △HEB ≅ △HCF
(ii) GEHF is a parallelogram.

Mathematics

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

Mid-point Theorem

Related Questions

In the adjoining figure, ABCD is a parallelogram. E and F are mid-points of the sides AB and CD respectively. The straight lines AF and BF meet the straight lines ED and EC in points G and H respectively. Prove that(i) △HEB ≅ △HCF(ii) GEHF is a parallelogram.

In the adjoining figure, ABCD is a parallelogram. E and F are mid-points of the sides AB and CD respectively. The straight lines AF and BF meet the straight lines ED and EC in points G and H respectively. Prove that
(i) △HEB ≅ △HCF
(ii) GEHF is a parallelogram.