The Tangent Object Snap lets you snap a line at a tangent point on another curve. The tangent is the point where two curves touch at exactly one point, and this is what the Tangent Object Snap does; it lets you snap to the tangent points of two curves. To use this Object Snap, once again make sure it is active in the status bar:

Make a circle and select the Line command and click on any point near the circle and then move your cursor close to the circle. You will notice a green Tangent Object Snap icon, like the one shown here:

Click to snap the line at the tangent point of the circle and you will have a line that starts at a random point but ends at a tangent point on the circle.