In triangle ABC, M is the midpoint of BC, AB=12, and AC=16. Points E and F are taken on AC and AB respectively, and EF and AM intersect at G. If AE=2AF then what is EG/GF?
+26367 In triangle ABC,
M is the midpoint of BC, AB=12, and AC=16.
Points E and F are taken on AC and AB respectively, and EF and AM intersect at G.
If AE=2AF then what is EG/GF?
\(\text{Let $\angle EAG=\epsilon_1$ } \\ \text{Let $\angle AMC=\epsilon_2$ } \\ \text{Let $\angle AGE=\epsilon_3$ } \\ \text{Let $\angle GAF=\delta_1$ } \\ \text{Let $\angle AMB=\delta_2=180^\circ-\epsilon_2 $ } \\ \text{Let $\angle AGF=\delta_3=180^\circ-\epsilon_3 $ } \\ \text{Let $EG=\mathbf{x}$ } \\ \text{Let $GF=\mathbf{y}$ } \\ \text{Let $AF=f$ } \\ \text{Let $AE=2AF=2f$ } \)
1. sin-rule
\(\begin{array}{|lrcll|} \hline (1) & \dfrac{\sin(\epsilon_1)}{\dfrac{CB}{2}} &=& \dfrac{\sin(\epsilon_2)}{16} \\\\ \hline & \dfrac{\sin(\delta_1)}{\dfrac{CB}{2}} &=& \dfrac{\sin(180^\circ-\epsilon_2)}{12} \\\\ (2) & \dfrac{\sin(\delta_1)}{\dfrac{CB}{2}} &=& \dfrac{\sin(\epsilon_2)}{12} \\\\ \hline & \dfrac{CB}{2}\sin(\epsilon_2) = 16\sin(\epsilon_1)&=& 12\sin(\delta_1) \\ & 16\sin(\epsilon_1)&=& 12\sin(\delta_1) \\\\ & \dfrac{\sin(\epsilon_1)}{\sin(\delta_1)} &=& \dfrac{12}{16} \\\\ (3) & \mathbf{ \dfrac{\sin(\epsilon_1)}{\sin(\delta_1)} } &=& \mathbf{ \dfrac{3}{4} } \\ \hline \end{array} \)
2. sin-rule
\(\begin{array}{|lrcll|} \hline (4) & \dfrac{\sin(\epsilon_1)}{x} &=& \dfrac{\sin(\epsilon_3)}{2f} \\\\ \hline & \dfrac{\sin(\delta_1)}{y} &=& \dfrac{\sin(180^\circ-\epsilon_3)}{f} \\\\ (5) & \dfrac{\sin(\delta_1)}{y} &=& \dfrac{\sin(\epsilon_3)}{f} \\\\ \hline & \dfrac{\sin(\epsilon_3)}{f} = \dfrac{2\sin(\epsilon_1)}{x} &=& \dfrac{\sin(\delta_1)}{y} \\\\ &\dfrac{x}{y} &=& 2\cdot \dfrac{\sin(\epsilon_1)}{\sin(\delta_1)} \quad | \quad \dfrac{\sin(\epsilon_1)}{\sin(\delta_1)} = \dfrac{3}{4} \\\\ &\dfrac{x}{y} &=& 2\cdot \dfrac{3}{4} \\\\ & \mathbf{ \dfrac{x}{y} } &=& \mathbf{ \dfrac{3}{2} } \\ \hline \end{array}\)
\(\mathbf{\dfrac{EG}{GF} = \dfrac32}\)
