Intersecting Chords Theorem in Circle

Since $P$ is the midpoint of chord $AB$, $AP = PB = 24/2 = 12\text{ cm}$. $MN$ is perpendicular to $AB$ at $P$, so $AB$ and $MN$ are two intersecting chords at $P$. By the Intersecting Chords Theorem: $AP \times PB = PM \times PN$ $12 \times 12 = 8 \times PN$ $144 = 8 \times PN \implies PN = 18\text{ cm}$.