Parallelogram ABCD and rectangle ABEF are on the same base and have equal areas. Show the perimeter of parallelogram is greater than rectangle.

given: ABCD is a rectangle and EFCD is a parallelogram, with the same base CD and have equal areas.

TPT: perimeter of the parallelogram EFCD > perimeter of the rectangle

proof:

since parallelogram and rectangle have equal areas with same base CD, therefore it will be between same set of parallel lines.

CD=EF............(1) [opposite sides of the parallelogram]

CD=AB...........(2) [opposite sides of the rectangle]

from (1) and (2), EF=AB...........(3)

in the triangle DAE,

since ∠DAE=90 deg

ED>AD [since length of the hypotenuse is greater than other sides]..........(4)

CF>BC [since CF=ED and BC=AD]...............(5)

perimeter of parallelogram EFCD

=EF+FC+CD+DE

=AB+FC+CD+DE [using (3)]

>AB+BC+CD+AD [using (5)]

which is the perimeter of the rectangle ABCD

therefore perimeter of parallelogram with the same base and with equal areas is greater than the perimeter of the rectangle.

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