Find the area of the triangular field with the length of the sides $360 \, m$,$450 \, m$,and $450 \, m$.

State whether each of the following statements is true or false:
$(1)$ In a square,if the length of each side is $20 \, cm$,then the length of its diagonal is $20 \sqrt{2} \, cm$.
$(2)$ If each side of an equilateral triangle is doubled,the area of the triangle is twice the area of the original triangle.

Find the area of the equilateral triangle having length of each side $50\, cm$.

If the sides of a triangular field are $290 \, m$,$290 \, m$,and $400 \, m$,find the area of the field in $m^2$.

In a quadrilateral $ABCD$,$AB = 29 \, cm$,$BC = 35 \, cm$,$CD = 14 \, cm$,$DA = 50 \, cm$ and $AC = 48 \, cm$. Find its area. (in $, cm^2$)

The sides of a quadrilateral $ABCD$ are $6 \, cm$,$8 \, cm$,$12 \, cm$,and $14 \, cm$ (taken in order) respectively,and the angle between the first two sides is a right angle. Find its area in $cm^2$.