Circles inscribed in a rectangle are tangent at distinct points; find the radius of the smaller circle based on the dimensions of the rectangle. 
A rectangle of height 8 and width 6 contains four circles; two large and equivalent circles go about the height of the rectangle. Two smaller and equivalent circles are tangent to the rectangle at exactly one point, and are also tangent to the larger circles. What is the radius of the smaller circles? (Figure is not drawn to scale)
I have been stuck on this problem for a while. 

I first deduced that the radius of the larger circle is 2. I then connected the radii of the smaller circle to each radii of the larger circles. Therefore, I used the handy Pythagorean Theorem to solve for R. $(R+2)^2
+(R+2)^2=4^2$
I obtained a final answer of $2(\sqrt2-1)$, which apparently is fallacious. Why is my answer fallacious?

Although you are on the right track, you chose the wrong triangle. The one you chose is not a right triangle, which is why your answer is wrong.
The following works much more easily:

Side lengths here are $l_1 = 2+r$, $l_2=3-r$, and $l_3=2$, yielding the equation $(2+r)^2=(3-r)^2+4$ and in turn the answer, $r=\frac9{10}$.
