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

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

KnowledgeBoat · Accepted Answer

Given: ABCD is a square and E, F, G and H are midpoints of sides AB, BC, CD and DA respectively. To prove: The quadrilateral EFGH is a square. Construction: Join EF, FG, GH and HE to form quadrilateral EFGH. Also, draw the diagonals AC and BD of square ABCD. Proof: In Δ ACD, G and H are the midpoints of CD and AD. By the Midpoint Theorem: GH ∥ AC and GH = AC ...................(1) Similarly, in Δ ABC, E and F are the midpoints of AB and BC. By the Midpoint Theorem: EF ∥ AC and EF = AC ...................(2) From (1) and (2), we get: EF ∥ GH and EF = GH = AC ...................(3) Similarly, in Δ ABD, E and H are midpoints of AB and AD, so: EH ∥ BD and EH = BD ...................(4) In Δ BCD, G and F are midpoints of CD and BC, so: ∴ FG ∥ BD and FG = BD ...................(5) From (4) and (5), we get: EH ∥ FG and EH = FG = BD ...................(6) Since diagonals of a square are equal, we know: AC = BD Dividing both sides by 2, we get: Substituting above value in equation (3) and (6), EF = GH = EH = FG ...................(7) Since opposite sides are equal and parallel, EFGH is a parallelogram. In Δ GOH and Δ GOF, OH = OF (diagonals of parallelogram bisect each other) GH = GF (from (7)) OG = OG (Common side) By SSS Congruence, Δ GOH ≅ Δ GOF So, corresponding angles are equal: ∠GOH = ∠GOF (C.P.C.T.) Since GOH and GOF are on a straight line: ⇒ ∠GOH + ∠GOF = 180° ⇒ ∠GOH + ∠GOH = 180° ⇒ 2∠GOH = 180° ⇒ ∠GOH = ⇒ ∠GOH = 90° Thus, the diagonals of EFGH are perpendicular to each other, confirming that EFGH is a square. Hence, the quadrilateral formed by joining the mid-points of the adjacent sides of a square, is also a square.

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

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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

P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle.Under what condition will PQRS be a square ?

In a scalene triangle ABC, AD and BE are medians. F is a point on AC so that DF//BE. Show that : AC = 4 x EF.

ABC is an isosceles triangle with AB = AC = 13 cm and BC = 10 cm. Calculate the length of the perpendicular from A to BC.

In triangle ABC, AB > AC. E is the mid-point of BC and AD is perpendicular to BC. Prove that :AB2 + AC2 = 2AE2 + 2BE2

P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle.
Under what condition will PQRS be a square ?

In triangle ABC, AB > AC. E is the mid-point of BC and AD is perpendicular to BC. Prove that :
AB² + AC² = 2AE² + 2BE²