2012 AMC 10A Problem 15

We can use coordinate geometry to figure out where the intersection of the two lines occurs. Let $A$ be the origin and $B = (1, 0).$ Then the slope of the line through $A$ is $-\dfrac{1}{2},$ which makes the equation of the line $$y = -\dfrac{1}{2}x.$$ The slope of the line through $B$ is $2.$ The $y$-intercept is $-2.$ This makes the equation of this line $$y = 2x - 2.$$ Equating the equations, we get $$2x - 2 = -\dfrac{1}{2}x$$$$x = \dfrac{4}{5}.$$ This makes the $y$-coordinate of $C$ $$-\dfrac{1}{2} \cdot \dfrac{4}{5} = -\dfrac{2}{5}.$$ The area of triangle $ABC$ is then $$\dfrac{1}{2} \cdot 1 \cdot \dfrac{2}{5} = \dfrac{1}{5}.$$ Thus, B is the correct answer.