Q. 195.0( 3 Votes )

Parallelogram ABC

Answer :

Given.


Parallelogram ABCD of which midpoints are E, F, G and H of sides AB, BC, CD and DA respectively


Formula used.


SAS congruency rule = If 2 sides and angle between them are equal in both triangles then both triangles are congruent



In triangle DGH and triangle BEF


DG= EB AB= CD then their half also be equal


DH= FB BC= DA then their half also be equal


D= B opposite angles of parallelogram are equal


triangle DGH triangle BEF [By SAS congruency]


GH = EF


In triangle AEH and triangle CGF


GC= AE AB= CD then their half also be equal


AH= FC BC= DA then their half also be equal


C= A opposite angles of parallelogram are equal


triangle DGH triangle BEF [By SAS congruency]


HE = GF


If both pair of sides are equal in quadrilateral then this is called parallelogram


EFGH is a parallelogram


Join the mid-points FH


Parallelogram divides in 2 equal parallelograms


In parallelogram ABFH and triangle EFH


Both lies on same base FH


And both lies on parallel lines AB and FH


AB || FH opposite sides of parallelogram are parallel


Then


Area of triangle EFH = × Area of parallelogram ABFH


In parallelogram FHDC and triangle EFG


Both lies on same base FH


And both lies on parallel lines CD and FH


CD || FH opposite sides of parallelogram are parallel


Then


Area of triangle GFH = × Area of parallelogram FHDC


Adding both we get;


Area of [Δ EFG+Δ GFH]=×Area of parallelogram [ABFH+FHDC]


Area of parallelogram EFGH = ×Area of parallelogram ABCD


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