Let’s first draw the trapezium.
Let the distance between $\text{RS},$ and $\text{PQ}$ be $h\;\text{cm}.$
Here, $\text{AB = 6 cm} \Rightarrow {\color{Blue}{\boxed{\text{PA + BQ = 5 cm}}}} \quad \longrightarrow (1)$
The $\triangle \text{PAS, QBR}$ are right angle triangle, so we can apply the Pythagorean theorem.
Now, in $\triangle \text{PAS},$
${\color{Green}{\text{(Hypotenuse)}^{2} = \text{(Perpendicular)}^{2} + \text{(Base)}^{2}}}$
$\Rightarrow (\text{PS})^{2} = (\text{AS})^{2} + (\text{PA})^{2}$
$\Rightarrow 3^{2} = h^{2} + (\text{PA})^{2}$
$\Rightarrow h^{2} + (\text{PA})^{2} = 9 \quad \longrightarrow (2)$
Now, in $\triangle \text{PAS},$
$\Rightarrow (\text{QR})^{2} = (\text{RB})^{2} + (\text{BQ})^{2}$
$\Rightarrow 4^{2} = h^{2} + (\text{BQ})^{2}$
$\Rightarrow h^{2} + (\text{BQ})^{2} = 16 \quad \longrightarrow (3)$
Subtract equation $(2)-(3).$
$\qquad \require{cancel}\begin{array}{} {\color{Red}{\cancel{h^{2}}}} + (\text{PA})^{2} = 9 \\ {\color{Red}{\cancel{h^{2}}}} + (\text{BQ})^{2} = 16 \\\; – \qquad \;\; – \qquad \quad – \\\hline (\text{PA})^{2} – (\text{BQ})^{2} = -7 \end{array}$
$\Rightarrow (\text{PA – BQ}) (\text{PA + BQ}) = -7 \quad [{\color{Green}{\because a^{2} – b^{2} = (a+b)(a-b)}}]$
$\Rightarrow (\text{PA – BQ}) \cdot 5 = -7$
$\Rightarrow {\color{Blue}{\boxed{\text{PA – BQ} = \frac{-7}{5}\;\text{cm}}}}\quad \longrightarrow (4)$
Adding the equation $(1)\; \& \;(4),$ we get.
$\text{PA + BQ + PA – BQ} = 5 \;– \frac{7}{5} $
$\Rightarrow \text{2PA} = \frac{25-7}{5}$
$\Rightarrow \text{2PA} = \frac{18}{5}$
$\Rightarrow {\color{Blue}{\boxed{\text{PA} = \frac{9}{5}\;\text{cm}}}}$
Put the value of $\text{PA}$ in the equation $(1).$
$h^{2} + (\text{PA})^{2} = 9$
$\Rightarrow h^{2} + \left(\frac{9}{5}\right)^{2} = 9$
$\Rightarrow h^{2} + \frac{81}{25} = 9$
$\Rightarrow h^{2} = 9 – \frac{81}{25}$
$\Rightarrow h^{2} = \frac{225-81}{25}$
$\Rightarrow h^{2} = \frac{144}{25}$
$\Rightarrow h = \sqrt{\frac{144}{25}}$
$\Rightarrow h = \frac{12}{5} = 2.40\;\text{cm}$
$\therefore$ The shortest distance between $\text{PQ}$ and $\text{SR (in cm)}$ is $2.40\;\text{cm}.$
Correct Answer $:\text{B}$