A line passes through the origin and has a slope of .
Which of the following points does the line pass through?
( 1/2 , 1)
(1, 2)
(2, 1)
(0, 1/2 )
We can use the point-slope form of a line to determine which of these points the line passes through:
The point-slope form of a line is given by $y - y_1 = m(x - x_1),$ where $(x_1, y_1)$ is a point on the line and $m$ is the slope.
Since the line passes through the origin, the point $(0,0)$ satisfies the equation of the line: $y - 0 = m(x - 0),$ or $y = mx.$
We can now substitute the coordinates of each of the given points into this equation and see which satisfies the equation:
If we substitute $(1/2, 1),$ we get $1 = m(1/2),$ or $m = 2.$ But the slope of the line is given to be $m,$ so this point does not satisfy the equation.
If we substitute $(1,2),$ we get $2 = m(1),$ or $m = 2.$ Notice that this is consistent with the slope given in the problem. So this point does satisfy the equation.
If we substitute $(2,1),$ we get $1 = m(2),$ or $2m = 1,$ or $m = \frac{1}{2}.$ But the slope of the line is given to be $m,$ so this point does not satisfy the equation.
If we substitute $(0,1/2),$ we get $1/2 = m(0),$ or $m = 0.$ But the slope of the line is given to be $m,$ so this point does not satisfy the equation.
Therefore, the only point that satisfies the equation $y = mx$ is $\boxed{(1,2)}.$