Question

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

Answer 1

Data: Parallelogram $ABCD$ and rectangle $ABEF$ are on the same base $AB$ and have equal areas.
To Prove: Perimeter of the parallelogram is greater than that of the rectangle.
Proof: parallelogram$ABCD$ and rectangle $ABEF$ are on same base $AB$ and in between $AB||FC$.
$AB=CD$ (Opposite sides of $\square$)
$AB=EF$ (Opposite sides of rectangle)
$\therefore CD=EF$
$\therefore AB+CD=AB+EF......\left(i\right)$
Perpendicular drawn from the vertex to base is only smaller than remaining lines.
$\therefore FA<AD$ and $BE<BC$
$\therefore AF+BE<AD+BC.....\left(ii\right)$
Comparing (i) and (ii),
$(AB+CD+AF+BE)<(AB+EF+AD+BC)$
$\therefore$ Perimeter of the parallelogram is greater than that of the rectangle.