Similar triangles

The triangles ABC and XYZ are similar. Find the missing lengths of the sides of the triangles.
a) a = 5 cm b = 8 cm
x = 7.5 cm z = 9 cm

b) a = 9 cm c = 12 cm
y = 10 cm z = 8 cm

c) b = 4 cm c = 8 cm
x = 4.5 cm z = 6 cm

Correct result:

c₁ =  6 cm
y₁ =  12 cm
a₂ =  6 cm
y₂ =  3 cm
a₃ =  6 cm
y₃ =  3 cm

Solution:

$$\begin{gathered} a_1=5\text{ cm } \\ {b}_1=8\text{ cm } \\ {x}_1=7.5\text{ cm } \\ {z}_1=9\text{ cm } \\ {k}_1=x_1\mathrm{/}{a}_1=7.5\mathrm{/}5={\displaystyle \frac{3}{2}}=1.5 \\ {k}_1=z_1\mathrm{/}{c}_1 \\ {c}_1=z_1\mathrm{/}{k}_1=9\mathrm{/}1.5=6\text{ cm} \end{gathered}$$

$$\begin{gathered} k_1=y_1\mathrm{/}{b}_1 \\ {y}_1=k_1\cdot b_1=1.5\cdot 8=12\text{ cm} \end{gathered}$$

$$\begin{gathered} b_2=4\text{ cm } \\ {c}_2=8\text{ cm } \\ {x}_2=4.5\text{ cm } \\ {z}_2=6\text{ cm } \\ {k}_2=z_2\mathrm{/}{c}_2=6\mathrm{/}8={\displaystyle \frac{3}{4}}=0.75 \\ {k}_2=x_2\mathrm{/}{a}_2 \\ {a}_2=x_2\mathrm{/}{k}_2=4.5\mathrm{/}0.75=6\text{ cm} \end{gathered}$$

$$\begin{gathered} k_2=y_2\mathrm{/}{b}_2 \\ {y}_2=k_2\cdot b_2=0.75\cdot 4=3\text{ cm} \end{gathered}$$

$$\begin{gathered} b_3=4\text{ cm } \\ {c}_3=8\text{ cm } \\ {x}_3=4.5\text{ cm } \\ {z}_3=6\text{ cm } \\ {k}_3=z_3\mathrm{/}{c}_3=6\mathrm{/}8={\displaystyle \frac{3}{4}}=0.75 \\ {k}_3=x_3\mathrm{/}{a}_3 \\ {a}_3=x_3\mathrm{/}{k}_3=4.5\mathrm{/}0.75=6\text{ cm} \end{gathered}$$

$$\begin{gathered} k_3=y_3\mathrm{/}{b}_3 \\ {y}_3=k_3\cdot b_3=0.75\cdot 4=3\text{ cm} \end{gathered}$$

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