Parallelogram ABCD and rectangle ABEF are on the same base and have equal areas Show the perimeter of parallelogram - Maths - Areas of Parallelograms and Triangles

Question

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

Ajanta Trivedi · Accepted Answer

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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 âˆ 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

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