Why aren't all points on a rolling ball moving? 
If a ball is rolling down a hill as shown, what can be said about the points indicated at that particular point in motion?

(A) Point A is moving to the left, Point B is at temporarily at rest, Point C is moving to the left
(B) Point A is moving vertically upward, Point B is moving to the left, Point C is moving vertically downward
(C) Point A is moving vertically upward, Point B is at temporarily at rest, Point C is moving vertically downward

The answer is C, but I don't understand why it is. Wouldn't rotational inertia mean that all the points are still moving?
 A: For pure rolling (no slipping) the ball is rotating about point B (the contact point). Thus A is moving up and a little to the right, and C downwards and a little to the left, whilest point B is at rest. If B was moving up or down it would break the contact, and if it move left or right it would slip.
Why?
The picture below shows the velocity vectors (blue) of points A and C as the ball rolls

The perpendicular to AB is along AB' according to geometry (lines from diametrical points meet pependicularly). Similarly for BC and B'C
So the answer is neither of the three actually. This drives the point that physics should be taught at the same time as geometry since they are both actually quite interconnected.
A: There are 2 kinds of motion here: translational  and rotational. 
The net velocity at any point is calculated by the vector sum of these two velocities.
For instance at point B, $v-rw=0$ (pure rolling condition).
That's why this point stays at rest.
A: Additional answer: look up "cycloid" via https://en.wikipedia.org/wiki/Cycloid.  There is a very good description (both pictorially and mathematically) of the path traced out by point B as the ball is rolling along.
Since my last posting, questions have arisen in the comments regarding how point B could be at rest while the ball is rotating.  Please see Figure 9-31, which shows a small spool rolling down a ruler.  Since there are white dots on a black spool, and the exposure time was relatively long, you can clearly see motion, as represented by the amount of blurring of the dots.  Note that the points at the top of the spool are moving at 2v, the center of mass is moving at v, and the bottom of the spool is momentarily at rest with respect to the ruler.

Image source: "Physics For Scientists And Engineers", Paul A. Tipler and Gene Mosca, 6th Edition, W. H. Freeman and Company, New York, 2008, p. 310.
A: What point are you turning about? 
That's right, point $B$! So surely this point will not move, just like when I lift my bike and spin the wheel, the center of the wheel doesn't move. 
When this is so, it should also be clear that the straight-line distance to $C$ cannot change. So for $C$ to reach the ground, it must must go downwards and slightly right (so little right, that it doesn't count here) 
