Step 1

We will use similarity of the given two triangles to find the value of x.

Step 2

\(\displaystyle\angle{B}{A}{E}\stackrel{\sim}{=}\angle{C}{A}{D}\) (same angle)

\(\displaystyle\angle{B}{E}{A}\stackrel{\sim}{=}\angle{C}{D}{A}\) (\(\displaystyle{90}^{\circ}\) both)

Hence \(\displaystyle\triangle{A}{E}{B}\sim\triangle{A}{D}{C}\)

Hence \(\displaystyle\frac{{{B}{E}}}{{{C}{D}}}=\frac{{{A}{B}}}{{{A}{C}}}\)

\(\displaystyle\Rightarrow\frac{{x}}{{15}}=\frac{{12}}{{{12}+{6}}}\)

\(\displaystyle\Rightarrow\frac{{x}}{{15}}=\frac{{12}}{{18}}\)

\(\displaystyle\Rightarrow{x}=\frac{{{12}\times{15}}}{{18}}\)

\(=10\)

Hence \(x = 10 cm\)