1. **Problem statement:** Given a circle with tangent segments VQ and VK, and a line segment IK, where $QV=8$ cm, $IK=24$ cm, and $KV=x$ cm, find the value of $x$. 2. **Key property:** Tangents drawn from an external point to a circle are equal in length. This means $QV = VK$. 3. **Apply the property:** Since $QV=8$ cm, then $VK = 8$ cm. 4. **Use the given segment IK:** The segment $IK$ is given as 24 cm, but it does not affect the equality of the tangent segments. 5. **Conclusion:** The unknown length $x = KV = 8$ cm. **Final answer:** $$x = 8 \text{ cm}$$