Tübitak Lise 1. Aşama 2023 Soru 25

[matematikolimpiyati]

$m(\widehat{BAC})=90^{\circ}$ olan bir $ABC$ üçgeninde $A$ noktasından $BC$ kenarına inen dikmenin ayağı $D$ ve $[AD]$ nin orta noktası $E$ olsun. $m(\widehat{BEC})=120^{\circ}$ ise $\dfrac{|BC|}{|AD|}$ kaçtır?

$\textbf{a)}\ 2  \qquad\textbf{b)}\ \dfrac{3\sqrt3}{2}  \qquad\textbf{c)}\ 3  \qquad\textbf{d)}\ 2\sqrt3  \qquad\textbf{e)}\ 4$

[geo]

Yanıt: $\boxed B$

$\tan \angle BED = \dfrac{BD}{DE} = \dfrac{2\cdot BD}{AD}$

$\tan \angle CED = \dfrac{CD}{DE} = \dfrac{2\cdot CD}{AD}$

$\begin{array}{lcl}
\tan 120^\circ &=& \tan (\angle BED + \tan \angle CED) \\ &=& \dfrac{\tan \angle BED + \tan \angle CED}{1-\tan \angle BED \cdot \tan \angle CED} \\ \\
 &=& \dfrac{\dfrac{2(BD+CD)}{AD}}{1-\dfrac{4\cdot BD\cdot CD}{AD^2}} \\ \\
&=& \dfrac{\dfrac{2\cdot BC}{AD}}{-3} \\ \\
&=&-\sqrt 3
\end{array}$

$\dfrac{BC}{AD}=\dfrac{3\sqrt 3}2$