In a rectangle $ABCD$ the diagonals intersect at $E$, and $F$ is on $BC$ with $BF = 2$ cm and $EF = 3$ cm. The length of diagonal $BD$ equals—

The diagonals of a rectangle bisect each other, so $E$ is the centre and $F$ is the midpoint of $BC$; thus $BC = 2\,BF = 4$ cm and $EF$ joins the centre to the side, equalling half of $AB$, so $AB = 6$ cm. Therefore $BD = \sqrt{AB^2 + BC^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}$ cm.