1. **Problem statement:** Given a circle with points $P$, $V$, and $N$ on the circumference and a point $Q$ outside the circle, we know the lengths $PV = 8$ cm, $NQ = 23$ cm, and we need to find $QV = x$ cm. 2. **Relevant theorem:** When two secants are drawn from a point outside a circle, the products of the lengths of the entire secant segments and their external parts are equal. This is called the Secant-Secant Power Theorem. 3. **Formula:** If $P$ and $N$ are points on the circle, and $Q$ is outside, then $$PV \times VQ = NQ \times QN_{ext}$$ where $VQ$ and $QN_{ext}$ are the external segments from $Q$ to the circle. 4. **Applying the theorem:** Here, $PV = 8$ cm, $NQ = 23$ cm, and $QV = x$ cm. Since $PV$ and $QV$ are parts of the same secant from $P$, and $NQ$ is the external segment from $N$, the theorem states: $$PV \times QV = NQ \times QV$$ But this is ambiguous; we need to clarify the segments. 5. **Clarification:** Usually, the theorem states: $$PQ \times QV = NQ \times QV$$ But since $PV$ and $NQ$ are given, and $QV$ is unknown, the problem likely implies: $$PV \times VQ = NQ \times QV$$ which simplifies to: $$8 \times x = 23 \times x$$ which is not possible unless $x=0$. 6. **Re-examining the problem:** The problem likely means the two secants intersect at $Q$, with segments $PV$ and $NQ$ given, and $QV$ unknown. The correct relation is: $$PV \times VQ = NQ \times QV$$ Since $PV = 8$ cm, $NQ = 23$ cm, and $QV = x$ cm, then: $$8 \times x = 23 \times x$$ which again is not solvable. 7. **Assuming a typo or missing segment:** If $PV$ and $NQ$ are external segments, and $QV$ is the internal segment, the theorem states: $$PV \times VQ = NQ \times QV$$ If $PV = 8$ cm, $NQ = 23$ cm, and $QV = x$ cm, then: $$8 \times x = 23 \times x$$ which is not possible. 8. **Conclusion:** Without additional information or clarification, the problem cannot be solved as stated. Please verify the problem statement or provide additional segment lengths. **Final answer:** Cannot determine $x$ with given information.