SparklingWater2

Because both circles are tangent to the same spot, the centers of the circles and the point of tangency form a line. There are two distances: one where both circles are on the same side, or where the circles are on seperate sides. 

If the two circles are on the same side, the distance would be 8 - 3 = $\boxed5$

If the two circles are on different sides, the distance would be 8 + 3 = $\boxed{11}$

The median of a trapezoid's length is the median of the two bases. So the median of ABCD is 12. Similarly, the median of ABMN's length is the median of the two bases AB and MN, so the length of PQ is $\boxed{8\frac12}$

If a and b must be positive integers, then there can be 29 ordered pairs :

1 + 29

2 + 28

...

28+2

29+1

There won't be 30 because 0 is not concidered positive or negative.

4x - 2y = 8 can be rearranged into point intercept form: y = 2x - 4

Parallel lines have the same slope, just different intercepts. The y intercept in the new equation is (0,-8) so the equation is $\boxed{y = 2x -8}$

Rotating 90$^\circ$ clockwise around the origin would make the new x coordinate the old y coordinate and the new y coordinate would be the negative of the old x coordinate so the coordinates of Z' are (-2,3)

D

There's a square root on the denominator of the solution.

multiply top and bottom by $\sqrt3$

2 or 6,8,2 or 6, 1/2 or 2

here are a few that are a little similar:

89+0$\cdot$(1+2+3+5+6)+4+7=100
89+0$\cdot$(1+2+3+4+7)+5+6=100

78+9$\cdot$2+3+1+0$\cdot$(4+5+6)=100
 

you can make some more off of that.