The parallel sides of a trapezoid are $ 3$ and $ 9$. The non-parallel sides are $ 4$ and $ 6$. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is:
[asy]defaultpen(linewidth(.8pt));
unitsize(2cm);

pair A = origin;
pair B = (2.25,0);
pair C = (2,1);
pair D = (1,1);
pair E = waypoint(A--D,0.25);
pair F = waypoint(B--C,0.25);

draw(A--B--C--D--cycle);
draw(E--F);

label("6",midpoint(A--D),NW);
label("3",midpoint(C--D),N);
label("4",midpoint(C--B),NE);
label("9",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 4: 3\qquad \textbf{(B)}\ 3: 2\qquad \textbf{(C)}\ 4: 1\qquad \textbf{(D)}\ 3: 1\qquad \textbf{(E)}\ 6: 1$
1. **Label the sides and set up the problem:**
   - Let the parallel sides of the trapezoid be \( AB = 9 \) and \( CD = 3 \).
   - Let the non-parallel sides be \( AD = 6 \) and \( BC = 4 \).
   - Let the line parallel to the bases that divides the trapezoid into two trapezoids of equal perimeters be \( EF \), where \( E \) is on \( AD \) and \( F \) is on \( BC \).

2. **Define the segments:**
   - Let \( AE = a \) and \( ED = 6 - a \).
   - Let \( BF = b \) and \( FC = 4 - b \).
   - Let the length of \( EF \) be \( x \).

3. **Set up the perimeter equations:**
   - The perimeter of the upper trapezoid \( EDCF \) is \( 3 + x + (6 - a) + (4 - b) \).
   - The perimeter of the lower trapezoid \( ABEF \) is \( 9 + x + a + b \).

4. **Equate the perimeters:**
   \[
   3 + x + (6 - a) + (4 - b) = 9 + x + a + b
   \]
   Simplify the equation:
   \[
   3 + x + 6 - a + 4 - b = 9 + x + a + b
   \]
   \[
   13 - a - b = 9 + a + b
   \]
   \[
   13 - 9 = a + b + a + b
   \]
   \[
   4 = 2(a + b)
   \]
   \[
   a + b = 2
   \]

5. **Set up the ratio equation:**
   - According to the problem, the ratio in which each of the non-parallel sides is divided is the same. Therefore:
   \[
   \frac{6 - a}{a} = \frac{4 - b}{b}
   \]

6. **Solve for \( a \) and \( b \):**
   - From the ratio equation:
   \[
   \frac{6 - a}{a} = \frac{4 - b}{b}
   \]
   Cross-multiply to get:
   \[
   (6 - a)b = a(4 - b)
   \]
   \[
   6b - ab = 4a - ab
   \]
   \[
   6b = 4a
   \]
   \[
   \frac{a}{b} = \frac{6}{4} = \frac{3}{2}
   \]

7. **Express \( a \) in terms of \( b \):**
   \[
   a = \frac{3}{2}b
   \]
   Substitute \( a = \frac{3}{2}b \) into \( a + b = 2 \):
   \[
   \frac{3}{2}b + b = 2
   \]
   \[
   \frac{5}{2}b = 2
   \]
   \[
   b = \frac{4}{5}
   \]
   \[
   a = \frac{3}{2} \times \frac{4}{5} = \frac{6}{5}
   \]

8. **Verify the ratio:**
   - The ratio \( \frac{a}{b} = \frac{\frac{6}{5}}{\frac{4}{5}} = \frac{6}{4} = \frac{3}{2} \).

Thus, the ratio in which each of the non-parallel sides is divided is \( \boxed{3:2} \).