Q. 94.0( 18 Votes )

ABCD is a square.

Answer :

Given:- *ABCD is a square


*E, F, G and H are the mid points of AB, BC, CD and DA


respectively


*AE = BF = CG = DH


Formula used:- *Isosceles Δ property


If 2 sides of triangle are equal then


Corresponding angle will also be equal


*Properties of quadrilateral to be square


All sides are equal


All angles are 90°


Solution:-


In Δ AHE, Δ EBF, Δ FCG, Δ DHG


AE = BF = CG = DH [Given]


If;


AE=EB [E is midpoint of AB]


BF=FC [F is midpoint of BC]


CG=GD [G is midpoint of CD]


DH=HA [H is midpoint of DA]


AE = BF = CG = DH


On replacing every part we get;


EB=FC=GD=HA;


A+ B+ C+ D=90° [All angles of square are 90°]


Hence;


All triangles Δ AHE, Δ EBF, Δ FCG, Δ DHG


are congruent by SAS property


Δ AHE Δ EBF Δ FCG Δ DHG


HE=EF=FG=GH [All triangles are congruent]


In Δ AHE, Δ EBF, Δ FCG, Δ DHG


all sides of square are equal and after the midpoint of each sides


Every half side of square are equal to half of other sides.


HA=AE , EB=FB ,FC=GC ,HD=DG


All Δ AHE, Δ EBF, Δ FCG, Δ DHG are isosceles


as central angle of all triangle is 90°


It makes all Δ AHE, Δ EBF, Δ FCG, Δ DHG are right angle isosceles Δ


all corresponding angles of equal side will be 45°


AHE= BEF= CFG= DHG= AEH= BFE= CGF= DGH=45°


as AB is straight line


Then; AEH+ HEF+ BEF=180°


45°+ HEF+45° =180°


HEF=180° -90° =90°


Similarly ;


EFG=90°


FGH=90°


GHE=90°


Conclusion:-


All angles are 90° and all sides are equal of quadrilateral


Hence quadrilateral is square


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