Proving Relationships of Parallel and Perpendicular Lines

The given line $y=\frac{1}{2}x+3,$ which is written in slope-intercept form, has a slope of $\frac{1}{2}.$ Let's remember the definition of the slope of a line. $$\begin{array}{c}m=\frac{\text{rise}}{\text{run}}\end{array}$$ For a slope of $\frac{1}{2},$ for every $2$ units we move right the line moves up $1$ unit. Recall that the slopes of perpendicular lines are opposite reciprocals. Therefore, a line which is perpendicular to the given line has a slope of $\text{-}2.$ We will look for lines such that for every unit we move to the right, the line moves $2$ unit down.

From the diagram above, we can see that only line $b$ has a slope of $\text{-}2.$ Therefore, line $b$ is the only perpendicular line to $y=\frac{1}{2}x+3.$