Question

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

Short answer

The value of missing lengths are$\mathit{y}\mathbf{=}\mathbf{3}$ and $\mathit{z}\mathbf{=}\mathbf{6}$.

Step-by-step solution

Step 1. State the properties of ‘similarity of triangle’.

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

That is, if $△ABC~△XYZ$

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

Step 2. State the rule of ‘cross multiplication’.

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 $ad=cb$.

Step 3. Calculate the missing lengths.

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 $z=6$.