Since $P Q R S$ is a rectangle, then $P Q$ is perpendicular to $Q R$. Therefore, the area of $\triangle P Q R$ is $\frac{1}{2}(P Q)(Q R)=\frac{1}{2}(5)(3)=\frac{15}{2}$. Since $P T=T U=U R$, then the areas of $\triangle P T Q, \triangle T U Q$ and $\triangle U R Q$ are equal. Therefore, the area of $\triangle T U Q$ is $\frac{1}{3}\left(\frac{15}{2}\right)=\frac{5}{2}$. Similarly, the area of $\triangle T U S$ is $\frac{5}{2}$. The area of quadrilateral $S T Q U$ is the sum of the areas of $\triangle T U Q$ and $\triangle T U S$, or $\frac{5}{2}+\frac{5}{2}=5$.