Answer: \(\dfrac{A(\triangle MNT)}{A(\triangle QRS)} = \dfrac{25}{81}\)
Since \(\triangle MNT \sim \triangle QRS\), vertex T corresponds to vertex S, so the altitudes from T and S are corresponding altitudes.
In similar triangles, corresponding altitudes are in the same ratio as corresponding sides, so the ratio of areas equals the ratio of the squares of corresponding altitudes:
\[\frac{A(\triangle MNT)}{A(\triangle QRS)} = \frac{5^2}{9^2} = \frac{25}{81}\]