Question

abcd is a square . e,f,g and h are the points ab,bc,cd,and da respectively . such that ae=bf= cg=dh.prove that efgh is a square

Answer 1

Join the mid points of the spare and after that join the diagonals Now by using mid point theorem show that HE parrallel and equal to GH Similarly show the HG parrallel and equal to EF since diagonals of square are equal therefore it's adjacent side of the inner figure is also equal .

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Answer 2

Given AE=BF=CG=DH

⟹ So, EB=FC=GD=HA

In △s AEH and BFE,

AE=BF, AH=EB,

∠A=∠B (each ∠ = 90⁰)

∴ △AEH ≅ △BFE

⟹ EH=EF and ∠4= ∠2.

But ∠1 + ∠4 = 90⁰ ⟹ ∠1 + ∠2 = 90⁰

⟹ ∠HEF = 90⁰

And if ∠HEF = 90⁰ so, ∠EFG = 90⁰, ∠FGH = 90⁰ and ∠GHE = 90⁰.

Hence Proved.

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