For 3.9, the key here is that we want our entire path to start at the origin (point C in the diagram), fully travel around the circle, and come back to the origin.

IF our goal is just to get to the circle with a shortest distance, going from C to A is shorter. BUT if we're going around the circle, what do you do after A? Note the distance from C to A plus the distance from A to D (on the circle) is larger than the distance from C to D directly.

The idea then is that the tangent line is the "furthest" point on the circle that we can get to with a straight line. For example, if we tried to go to a point E further around the circle, the line CE would intersect the circle at another point, which doesn't work.

For 5.27, remember that an altitude is always perpendicular to the opposite base. Let's look at one example in more detail, the altitude through $A=(a,0)$ needs to be perpendicular to the line $BC$ (where $B=(0,b)$ and $C=(c,0)$). Note the slope of the line between points $B$ and $C$ is $\dfrac{-b}{c}$. Hence, we want a line with slope $\dfrac{c}{b}$ through the point $(a,0)$. This is$$y-0=\frac{c}{b}(x-a) \text{ or } y = \frac{c}{b}x - \frac{ac}{b}.$$Similar calculations give the other altitudes.

Hope this helps!