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.

(A) $(1)$ True: The diagonal $d$ of a square with side $a$ is given by $d = a \sqrt{2}$. For $a = 20 \, cm$,$d = 20 \sqrt{2} \, cm$. Thus,the statement is true.
$(2)$ False: The area of an equilateral triangle with side $a$ is $A = \frac{\sqrt{3}}{4} a^2$. If the side is doubled $(a' = 2a)$,the new area $A'$ becomes $A' = \frac{\sqrt{3}}{4} (2a)^2 = \frac{\sqrt{3}}{4} (4a^2) = 4 \times A$. The area becomes $4$ times the original area,not twice.