Rectangle by definition given as the quadrilateral in which opposite sides are equal and each angle is equal to 90 degree
Where as in case of trapezium any one pair of opposite sides is parallel.
Can we say rectangle as trapezium .
[Math] Are rectangles are trapezium
geometry
Related Solutions
Quadrilaterals are not rigid: you can fix a side and move one of the other vertices freely while respecting all side lengths. The area changes when you move the vertex.
(image from http://www.mathsisfun.com/definitions/rigid.html)
You can parametrize the trapezium via its height $h$. Then the longer side is $$ 2 \sqrt{k^2-h^2} + k $$ and its area is $$ h (\sqrt{k^2-h^2} + k)\ .$$ The derivative for $h$ is $$k - \frac{h^2}{\sqrt{-h^2 + k^2}} + \sqrt{-h^2 + k^2}\ ,$$ which is zero for $h=0$ or $h=\pm \frac{\sqrt{3}}{2}k$. $h=\frac{\sqrt{3}}{2}k$ gives the proposed optimum, which you can be sure about after also checking the border case $h=k$ (which gives an area of $k^2$).
Best Answer
In general by definition rectangles are not trapzeoids, only one pair of sides are parallel in a trapezoid. There are some definitions though which define parallelograms to be trapzeoids. It comes down to the definition given in class.