Find the missing length if$\mathbf{△}\mathit{A}\mathit{B}\mathit{C}\mathbf{~}\mathbf{△}\mathit{X}\mathit{Y}\mathit{Z}$.

If two triangles are similar, then the ratio of the corresponding sides are equal.

That is, if $△ABC~△XYZ$

Then, $\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}$

If two ratios are equal, then the numerator of first fraction is multiplied by the denominator of second fraction and the numerator of second fraction is multiplied by the denominator of first fraction.

That is, if $\frac{a}{b}=\frac{c}{d}$then $ad=cb$.

Observe the figure given below.

From the figure,

$AB=4,CB=3,AC=2,XY=z,YZ=4.5\&XZ=y$

As $△ABC~△XYZ$

$\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}\dots \left(1\right)$

Substitute the values $AB=4,CB=3,AC=2,XY=z,YZ=4.5\&XZ=yin\left(1\right)$

$\frac{4}{z}=\frac{3}{4.5}=\frac{2}{y}$

Consider $\frac{4}{z}=\frac{3}{4.5}$

$\begin{array}{l}\frac{4}{z}=\frac{3}{4.5}\\ 4\times 4.5=3\times z\left[\text{By\hspace{0.17em}\hspace{0.17em}cross\hspace{0.17em}\hspace{0.17em}multiplication}\right]\\ 18=3z\\ 3z=18\\ \frac{3z}{3}=\frac{18}{3}\\ z=6\end{array}$

Consider $\frac{3}{4.5}=\frac{2}{y}$

$\begin{array}{l}\frac{3}{4.5}=\frac{2}{y}\\ 3\times y=2\times 4.5\left[\text{By\hspace{0.17em}\hspace{0.17em}cross\hspace{0.17em}\hspace{0.17em}multiplication}\right]\\ 3y=9\\ \frac{3y}{3}=\frac{9}{3}\\ y=3\end{array}$

Therefore,$y=3$ and $z=6$.