In $\mathrm{Rt} \triangle A B C$, $\angle A=90^{\circ}$, the inscribed square $P Q R S$ has one side $R S$ on the hypotenuse $B C$, $P$ is on $A B$, and $Q$ is on $A C$. Given $|A B|=a(a>0, a$ is a constant), $\angle B=\theta$, when $\theta$ varies, the minimum value of $S_{\triangle A B C}$ : $S_{P Q R S}$ is $\qquad$
$46 \frac{9}{4}$. $\quad S_{\triangle A B C}=\frac{1}{2} a^{2} \tan \theta, S_{\text {PQRS }}=\frac{4 a^{2} \sin ^{2} \theta}{(\sin 2 \theta+2)^{2}}$. The ratio of the two $y=1+\frac{\sin ^{2} 2 \theta+4}{4 \sin 2 \theta}$. As a function of $x=\sin 2 \theta$, $y$ is decreasing on $[0,1]$. From this, find the minimum value $y_{\min }=\frac{9}{4}$.