1. **State the problem:** We have a right triangle $\triangle MNO$ with $\angle O = 90^\circ$, and side lengths $NM = 85$, $MO = 84$, and $ON = 13$. We need to find the ratio that represents $\tan(\angle M)$. 2. **Recall the tangent definition:** For an angle in a right triangle, $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$. Here, $\theta = \angle M$. 3. **Identify sides relative to $\angle M$:** - Opposite side to $\angle M$ is $ON = 13$. - Adjacent side to $\angle M$ is $MO = 84$. 4. **Write the tangent ratio:** $$\tan(\angle M) = \frac{ON}{MO} = \frac{13}{84}.$$ 5. **Check the hypotenuse:** $NM = 85$ confirms the Pythagorean triple $13^2 + 84^2 = 85^2$, so the side lengths are consistent. **Final answer:** $$\boxed{\tan(\angle M) = \frac{13}{84}}.$$