Question

If $ABCDEF$ is a regular hexagon with its center at $0$, then the value of $\overline{AB}+\overline{AC}+\overline{AD}+\overline{AE}+\overline{AF}$

• A. $5\overline{AO}$
• B. $6\overline{AO}$
• C. $4\overline{AO}$
• D. None of these

Answer 1

The correct option is B $6\overline{AO}$

If $\overline{AB}=\overline{a},\overline{BC}=\overline{b}$

then $\overline{AC}=\overline{a}+\overline{b}$

$\overline{AD}=\overline{2b}$,

$\overline{AE}=\overline{AD}+\overline{DE}=2\overline{b}-\overline{a}$ and
$\overline{AF}=\overline{AE}+\overline{EF}$

$=\overline{AE}+\overline{CB}=2\overline{b}-\overline{a}-\overline{b}=\overline{b}-\overline{a}$
$\therefore \overline{AB}+\overline{AC}+\overline{AD}+\overline{AE}+\overline{AF}$

$=6\overline{BC}$

$=6\overline{AO}$