#### Answer

measure of arc $BC=120^{\circ}$

#### Work Step by Step

The measure of an entire circle is $360^{\circ}$. If you add the three arcs together...$AB+BC+CA$ you have the entire circle. So...
arc $AB=2x$, arc $BC=3x$, and arc $AC=4x$.
$2x+3x+4x=360$
combine like terms...
$9x=360$
Divide both sides by 9...
$x=40$
Plug $x=40$ into the $3x$ (the measure of arc $BC$) and the measure of arc $BC=120^{\circ}$.