Area of a square $= (........... )^{2}.$

State whether each of the following statements is true or false:
$(1)$ $ABCD$ is a rhombus. If $AB = 12 \, \text{cm}$, then the perimeter of $ABCD$ is $48 \, \text{cm}$.
$(2)$ $PQRS$ is a square. If $PR = 8 \, \text{cm}$, then the perimeter of $PQRS$ is $32 \, \text{cm}$.

$A$ rectangular plot is given for constructing a house, having a measurement of $40 \, m$ long and $15 \, m$ in the front. According to the laws, a minimum of $3 \, m$ wide space should be left in the front and back each, and $2 \, m$ wide space on each of the other sides. Find the largest area where the house can be constructed (in $m^2$).

$ABCD$ is a rectangle. If $AB = 12 \, cm$ and $BC = 7 \, cm$,then find the area of $ABCD$ in $cm^2$.

Can all the four angles of a quadrilateral be obtuse angles? Give reason for your answer.

For any convex quadrilateral $ABCD$,prove that $AB + BC + CD + DA > AC + BD$.