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# Help pls

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In triangle $ABC$, points $D$ and $F$ are on $\overline{AB},$ and $E$ is on $\overline{AC}$ such that $\overline{DE}\parallel \overline{BC}$ and $\overline{EF}\parallel \overline{CD}$. If $AF = 9$ and $DF = 3$, then what is $BD$?

May 19, 2022

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Help pls

$$\text{In triangle ABC, points D and F are on \overline{AB},\\ and E is on \overline{AC} such that \overline{DE}\parallel \overline{BC} and \overline{EF}\parallel \overline{CD}.\\ If AF = 9 and DF = 3, then what is BD?}$$

$$\begin{array}{|lrcll|} \hline (1) & \dfrac{\overline{AC}}{\overline{AE}} &=& \dfrac{9+3}{9} \\\\ (2) & \dfrac{\overline{AC}}{\overline{AE}} &=& \dfrac{9+3+BD}{9+3} \\ \hline \\ (1)=(2): &\dfrac{\overline{AC}}{\overline{AE}}= \dfrac{9+3}{9} &=& \dfrac{9+3+BD}{9+3} \\\\ & \dfrac{9+3}{9} &=& \dfrac{9+3+BD}{9+3} \\\\ & \dfrac{12}{9} &=& \dfrac{12+BD}{12} \\\\ & \dfrac{4}{3} &=& 1+\dfrac{BD}{12} \\\\ & \dfrac{BD}{12}&=& \dfrac{4}{3}-1 \\\\ & \dfrac{BD}{12}&=& \dfrac{1}{3}\\\\ & BD &=& \dfrac{12}{3} \\\\ & \mathbf{BD} &=& \mathbf{4} \\ \hline \end{array}$$

May 20, 2022