In the figure below, $\overline{M P}$ and $\overline{N Q}$ intersect at $O, N O=16 \mathrm{~cm}, M N=12 \mathrm{~cm}, M P=25 \mathrm{~cm}$, and $Q O=4 \mathrm{~cm}$. Find the length of $\mathrm{cm}$

Step 1 ：Given that $\overline{M P}$ and $\overline{N Q}$ intersect at $O$, with $N O=16 \mathrm{~cm}$, $M N=12 \mathrm{~cm}$, $M P=25 \mathrm{~cm}$, and $Q O=4 \mathrm{~cm}$.

Step 2 ：From the given information, we can see that $\triangle MNO$ and $\triangle QPO$ are similar by the Angle-Angle (AA) criterion (since they share an angle at $O$ and $\angle MNO = \angle QPO$ because they are corresponding angles of parallel lines).

Step 3 ：Therefore, the ratios of their corresponding sides are equal. We can set up the proportion $\frac{NO}{MO} = \frac{OQ}{OP}$ and solve for $OP$.

Step 4 ：Substituting the given values into the proportion, we get $\frac{16}{12} = \frac{4}{OP}$.

Step 5 ：Solving for $OP$, we find that $OP = 5.0 \mathrm{~cm}$.

Step 6 ：Finally, we can find the length of $\overline{PQ}$ by subtracting $OP$ from $MP$, which gives us $PQ = MP - OP = 25 - 5 = 20 \mathrm{~cm}$.

Step 7 ：Final Answer: The length of $\overline{PQ}$ is $\boxed{20}$ cm.