Let $x$ be the side of the larger square and $y$ be the side of the smaller square.
According to the question$:$
- $y=x-5\quad \rightarrow(1)$
- $x^2 =4y^2\rightarrow(2)$
Putting $(1)$ in $(2)$ we get
$x^2 = 4(x-5)^2$
$\implies x^{2}=4(x^{2}+25-10x)$
$\implies 3x^{2}-40x+100=0$
$\implies 3x^{2}-30x-10x+100=0$
$\implies 3x(x-10)-10(x-10)=0$
$\implies (x-10)(3x-10)=0$
$\implies x=10 \ cm \ \ (or) \ x = \dfrac{10}{3} \ cm$
So, the correct answer is $(C).$