Question

$ABC$ is an equilateral triangle with side length $k$ such that the vertices $B$ and $C$ lie on two parallel lines at a distance of $9$ units. If $A$ lies between the parallel lines at a distance of $5$ units from one of them, then the value of $\left[k\right]$ is
($[.]$ denotes the greatest integer function)

• A. $7$
• B. $8$
• C. $9$
• D. $10$

Answer 1

The correct option is C $9$


From diagram,
$\cos \theta =\frac{4}{k},\sin \theta =\frac{\sqrt{k^2-16}}{k}$
$\cos ({120}^{\circ }-\theta )=\frac{5}{k}\Rightarrow \cos {120}^{\circ }\cos \theta +\sin {120}^{\circ }\sin \theta =\frac{5}{k}\Rightarrow -\frac{1}{2}\times \frac{4}{k}+\frac{\sqrt{3}}{2}\times \frac{\sqrt{k^2-16}}{k}=\frac{5}{k}\Rightarrow \frac{\sqrt{3}}{2}\times \frac{\sqrt{k^2-16}}{k}=\frac{7}{k}\Rightarrow \frac{\sqrt{k^2-16}}{k}=\frac{14}{k\sqrt{3}}\Rightarrow 3k^2-48=196\Rightarrow k=\sqrt{\frac{244}{3}}=\sqrt{81.33}\therefore \left[k\right]=9$