Define variables and set up equations using the Pythagorean theorem.

Let $r$ be the radius of the circle.
Let $d_1$ be the perpendicular distance from the center to the chord of length 4 cm.
Let $d_2$ be the perpendicular distance from the center to the chord of length 6 cm.
A perpendicular from the center to a chord bisects the chord.
For the 4 cm chord, its half-length is $\frac{4}{2} = 2$ cm.
For the 6 cm chord, its half-length is $\frac{6}{2} = 3$ cm.
Using the Pythagorean theorem, we can write two equations:

The chords are on the same side of the center, and the distance between them is 2 cm. Since the longer chord is closer to the center, $d_1$ must be greater than $d_2$.
Therefore, the distance between the chords is $d_1 - d_2 = 2$ cm.
This implies $d_1 = d_2 + 2$.

Substitute $d_1 = d_2 + 2$ into Equation (1):

Now, equate Equation (2) and Equation (3) since both are equal to $r^2$:

Subtract $d_2^2$ from both sides:

Subtract 8 from both sides:

Substitute the value of $d_2$ into Equation (2):

To find $r$, take the square root of 9.0625:

The radius of the circle is $\frac{\sqrt{145}}{4} \text{ cm}$.