Math Reasonings Solved Exercises

Exercise 52

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 \( x \)?

  1. \( \frac{1}{4} \)

  2. \( \frac{2}{3} \)

  3. \( 3 \)

  4. \(4\)

  5. \( \frac{3}{2} \)

Try to solve it before checking the answer.

  1. \(4\)

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,

\[ \begin{aligned} \frac{2}{3} = \frac{ x – 2 }{ 3x – 9 } &\Rightarrow 2(3x – 9) = 3(x – 2) \\[1em] &\Rightarrow 6x – 18 = 3x – 6 \\[1em] &\Rightarrow 3x = 12 \Rightarrow x = \boldsymbol{4} \end{aligned} \]

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