Alan

Here's the equation of the curve:

Can you take it from here?

"P= (-9,8)       PR has the length of 20 units

A=(3,-1) 

find the nearest distance between A and PR"

Because the position of R is not specified I take this to mean "find the nearest distance between A and the locus of R".   

"The height of a baseball t seconds after it is hit up in the air is given by h=-9t2+52t+3. What is the maximum height of the ball?

I also need to know:

What is the y-intercept of the ball's path in the problem above AND what is its meaning in the problem?

How long was the ball in the air in the problem above?"

It always looks easy in retrospect! I looked at it in other, less productive ways first before I came up with the above.

Here's my take on this:

I guess what is wanted is that a = -1, b = 5 and c = 2.  However, this is a bit silly, as r can be expressed in an infinite number of ways simply by multiplying numerator and denominator by any positive number, p.  Then we would have a = -p, b = 5p² and c = 2p.

Having chosen the first vertex at random there are 9 others that can be reached from it with a line along a surface.  Hence there are 19 - 9 = 10 others for which the line must go inside the dodecahedron.  The probability is therefore 10/19.

The trick is to find the right way to look at the problem. It took me a while to find the right way!

As follows:

I'm not so sure about this Chris.  In statement: "Then by SAS , triangle ABE  is congruent to triangle CBE" you have assumed that the bisector of ABC meets the bisector of BCD at the same point on AD (and you then proceed to show everything is consistent if this is the case).  In other words, you have assumed what you set out to prove!  I think you should start with the assumption that the two bisectors meet at potentially different points, say E and F, on AD, and then show that E and F must be the same point.  

Should this be:

   ? - 3 = ?

-(6 -  2 = 4)

---------------

  ? -  1 = ?

If so, rewrite as:

  a - 3 = b

-(6 - 2 = 4)

--------------

  c - 1 = d

In this case there are an infinite number of solutions!

We have four equations:

a - 3 = b        

a - 6 = c      

c - 1 = d       

b - 4 = d     

These are satisfied by

a = 7 + p,   b = 4 + p,   c = 1 + p,   d = p  where p is any number

So for example, if p = 0, then a = 7, b = 4, c = 1 and d = 0

If p = 4, then a = 11, b = 8, c = 5 and d = 4

etc.