Answer
The value of a is $53.8$.
Work Step by Step
From the provided figure, we get
$\begin{align}
& \angle A=22{}^\circ +10{}^\circ \\
& =32{}^\circ
\end{align}$
Now to find the angle C, we will use the angle sum property of triangles:
$A+B+C=180{}^\circ $
So,
$\begin{align}
& A+B+C=180{}^\circ \\
& C=180{}^\circ -A-B \\
& C=180{}^\circ -32{}^\circ -90{}^\circ \\
& C=58{}^\circ
\end{align}$
By the linear pair of angles, we get
$\begin{align}
& \angle ADC+\angle ADB=180{}^\circ \\
& 100{}^\circ +\angle ADB=180{}^\circ \\
& \angle ADB=180{}^\circ -100{}^\circ \\
& \angle ADB=80{}^\circ
\end{align}$
Now, we will find AD by using the law of sines:
$\begin{align}
& \frac{AD}{\sin 90{}^\circ }=\frac{BA}{\sin 80{}^\circ } \\
& AD=\frac{120\sin 90{}^\circ }{\sin 80{}^\circ } \\
& AD=\frac{120}{0.984} \\
& AD=121.951
\end{align}$
Now, using the law of sines we will find a:
$\begin{align}
& \frac{a}{\sin 22{}^\circ }=\frac{121.951}{\sin 58{}^\circ } \\
& \frac{a}{\sin 22{}^\circ }=\frac{121.951}{\sin 58{}^\circ } \\
& a=\frac{121.951\sin 22{}^\circ }{\sin 58{}^\circ } \\
& a\approx 53.8
\end{align}$
