question_answer

ABC is a triangle, where\[BC=2AB,\]\[\angle B=\text{ }30{}^\circ \]and \[\angle A=90{}^\circ .\]The magnitude of the side AC is

A) \[\frac{2BC}{3}\]

B) \[\frac{3BC}{4}\]

C) \[\frac{BC}{\sqrt{3}}\]

D) \[\frac{\sqrt{3}BC}{2}\]

Correct Answer: D

Solution :

[d] Given that, \[\angle A=90{}^\circ \]and \[\angle B=30{}^\circ \] In \[\Delta ABC,\] \[\Rightarrow \]   \[\angle C=180{}^\circ -90{}^\circ -30{}^\circ \] \[\Rightarrow \]   \[\angle C=60{}^\circ \]and \[BC=2AB\] (i) From Pythagoras theorem, \[B{{C}^{2}}=A{{C}^{2}}+A{{B}^{2}}\] \[\Rightarrow \]   \[{{(2\,AB)}^{2}}=A{{C}^{2}}+A{{B}^{2}}\] [from Eq. (i)] \[\Rightarrow \]   \[AC=\frac{\sqrt{3}}{2}.BC\]