The ratio of length and breadth of a rectangle is $21:20$ and its perimeter is $164\text{cm}$. What is the length of its diagonal (in cm)?

Let the length ($l$) be $21x$ and breadth ($b$) be $20x$.Perimeter of a rectangle = $2(l+b)$ $
$2(21x+20x)=164$ $
$$2(41x)=164$$
$$82x=164⟹x=2$$
So,Length ($l$) = $21\times 2=42\text{cm}$ Breadth ($b$) = $20\times 2=40\text{cm}$ The diagonal ($d$) of a rectangle is given by:
$$d=\sqrt{l^2+b^2}$$
$$d=\sqrt{42^2+40^2}$$
$$d=\sqrt{1764+1600}$$
$$d=\sqrt{3364}=58\text{cm}$$