Tangent and Slope of Lines

### Slope of a Line Passing Through the Origin

If a line passes through the origin then the equation for that line will take the form:

(1)
$$y=mx$$

Slope is defined as:

(2)
\begin{align} slope=\frac{rise}{run}=\frac{y_2 - y_1}{x_2 - x_1} \end{align}

If we use the origin as one of those points we can see that:

(3)
\begin{align} slope=\frac{y}{x}=tan \theta \end{align}

The IB loves this detail. It shows up once every 3-4 years. The slope of of line that goes through the origin can be written:

(4)
\begin{align} y= tan(\theta) \cdot x \end{align}

The questions involving this idea are generally simple and mostly require a student to remember this or derive it, but frankly you don't have time to derive…