Given the information from the problem, we can establish that points K, L, and M form a vertical line where K is above the center F of the circle, and L and M are on the circumference of the circle.
Since KL and KM represent distances from point K to points L and M respectively, we can express the distances as follows:
- KL = distance from K to L = \( 3x + 2 \)
- KM = distance from K to M = \( 14 , \text{cm} \)
From the arrangement of points, we know that:
\[ KL + LM = KM \]
Since L and M are on opposite sides of point K, we can deduce that:
\[ LM = KL + KM \]
We can set up the equation based on the distances mentioned. Because LM is the distance between points L and M, we have:
\[ KL + LM = KM \]
Rearranging the equation to find the distances:
\[ LM = KM - KL \]
Substituting in the known values gives us:
\[ LM = 14 - (3x + 2) \]
\[ LM = 14 - 3x - 2 \]
\[ LM = 12 - 3x \]
Now looking at the entire distance LM and the fact it is equal to the difference of distances:
Since KL + LM = KM, we set those equal:
\[ (3x + 2) + (12 - 3x) = 14 \]
This simplifies and allows us to solve for x:
\[ 3x + 2 + 12 - 3x = 14 \]
The \(3x\) cancels out:
\[ 14 = 14 \]
This true statement does not help us find x directly.
Therefore, we solve for x by using the shorter method as KL and KM are both heights.
Thus, let’s solve for x using only KM = KL + LM where LM = 12 - 3x:
We know the circle's symmetry results in equal halves so:
Since total distance would calculate KL + LM as equal to the other side, it shows symmetry implies that:
So we need to find the discrepancy.
We adjust and state:
Since both total was equal to KL + 3x part equaling altogether:
We would say since 3x + 2 = 14 directly leads to:
\[ 3x + 2 = 14 \]
Moving 2 across gives:
\[ 3x = 12 \]
\[ x = 4 \]
Thus, the value of x is:
\[ \boxed{4} \]