A designer scales up a triangular motif so that the new motif is similar to the old one and covers 2.25 times as much area.
Find the ratio old length : new length for any pair of matching sides.

If the length ratio is 1 : $x$ then the area ratio is $1:x^2$, so $x^2=2.25$ and $x=1.5$, giving 1 : 1.5 (the same as 2 : 3). Quoting 1 : 2.25 uses the area factor as a length factor, 1 : 5.0625 squares a second time, and 1 : 1.125 halves 2.25 instead of taking its square root.