Answer
$B=46.5{}^\circ,a=7.69\text{ and }c=11.17$.
Work Step by Step
We know that the sum of a triangle's angles is:
$\begin{align}
& \angle A+\angle B+\angle C=180{}^\circ \\
& 43.5{}^\circ +\angle B+90{}^\circ =180{}^\circ \\
& \angle B=90{}^\circ -43.5{}^\circ \\
& \angle B=46.5{}^\circ
\end{align}$
Now, for the values of a and c:
$\begin{align}
& \tan \theta =\frac{BC}{AC}=\frac{a}{b} \\
& \tan 43.5{}^\circ =\frac{a}{8.1} \\
& a=8.1\times \tan 43.5{}^\circ \\
& a=7.68
\end{align}$
And by using the Pythagorean Theorem:
$\begin{align}
& {{a}^{2}}+{{b}^{2}}={{c}^{2}} \\
& {{c}^{2}}={{\left( 7.68 \right)}^{2}}+{{\left( 8.1 \right)}^{2}} \\
& c=11.16
\end{align}$
Thus, the values are:
$B=46.5{}^\circ,a=7.69\text{ and }c=11.17$.
