The symbol ∼ denotes similarity, that is, the triangles ABC and XYZ are similar. When two triangles are similar, they have the same shape but not necessarily the same size and their corresponding angles are the same. In turn, the corresponding sides of two corresponding triangles are in the same ratio. Graphically,
[tex]\frac{a}{a^{\prime}}=\frac{b}{b^{\prime}}=\frac{c}{c^{\prime}}[/tex]
So to find x in the small triangle, you have
[tex]\begin{gathered} \frac{15}{10}=\frac{9}{x} \\ \text{ Multiply by x on both sides of the equation} \\ \frac{15}{10}\cdot x=\frac{9}{x}\cdot x \\ \frac{15}{10}x=9 \\ \text{ Multiply by }\frac{10}{15}\text{ on both sides of the equation} \\ \frac{10}{15}\cdot\frac{15}{10}x=9\cdot\frac{10}{15} \\ x=6 \end{gathered}[/tex]
Finally, to find b in the large triangle, you have
[tex]\begin{gathered} \frac{15}{10}=\frac{b}{8} \\ \text{ Multiply by 8 on both sides of the equation} \\ \frac{15}{10}\cdot8=\frac{b}{8}\cdot8 \\ 12=b \end{gathered}[/tex]
Therefore, the lengths of the sides x and b are
[tex]\begin{gathered} x=6 \\ b=12 \end{gathered}[/tex]
