The triangles \( \triangle ABC \) and \( \triangle PBQ \) in the figure are similar triangles. Moreover, the value of the ratio \( \frac{ \overline{PB} }{ \overline{ AB}} \) is \( \frac{2}{3} \). What is the value of \( x \)?
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\( \frac{1}{4} \)
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\( \frac{2}{3} \)
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\( 3 \)
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\(4\)
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\( \frac{3}{2} \)
Try to solve it before checking the answer.
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\(4\)
Since they are similar triangles, we have
\[ \frac{\overline{PB}}{\overline{AB}} = \frac{\overline{PQ}}{\overline{AC}} \]But,
\[ \frac{\overline{PB}}{\overline{AB}} = \frac{2}{3} \hspace{1em} \text{ y } \hspace{1em} \frac{ \overline{PQ} }{ \overline{AC} } = \frac{ x – 2 }{ 3x – 9 } \]Hence,
\[ \begin{aligned} \frac{2}{3} = \frac{ x – 2 }{ 3x – 9 } &\Rightarrow 2(3x – 9) = 3(x – 2) \\[1em] &\Rightarrow 6x – 18 = 3x – 6 \\[1em] &\Rightarrow 3x = 12 \Rightarrow x = \boldsymbol{4} \end{aligned} \]