question_answer

From the point B a perpendicular BD is drawn on AC. If \[\cos 30=0.8\], find the length of AD.  

A) 45

B) 30

C) 50

D) 80

Correct Answer: A

Solution :

 In \[\Delta 's\] ABC and ABD, \[\angle A\] is common and \[\angle ABC=\angle ADB\] Hence \[\Delta \,ABC\] and \[\Delta \,ABD\] are equiangular \[\therefore \]  \[\frac{AD}{AB}=\frac{AB}{AC}\] or            \[AD=\frac{A{{B}^{2}}}{AC}=\frac{A{{C}^{2}}-B{{C}^{2}}}{AC}\]                                 \[=AC\left[ 1-{{\left( \frac{BC}{AC} \right)}^{2}} \right]\]                 \[=\frac{BC}{\cos {{30}^{o}}}[1-{{(\cos {{30}^{o}})}^{2}}]\]                 \[=\frac{100}{(0.8)}[1-{{(0.8)}^{2}}]\]                 \[=\frac{100\times 10}{8}\left[ 1-\frac{64}{100} \right]\]                 \[=\frac{100\times 10}{8}\times \frac{36}{100}=45\]