1. **Problem Statement:** Find the tangent of angle $\angle J$ in the right triangle with sides $KJ=12$, $KI=35$, and hypotenuse $IJ=37$. 2. **Formula:** The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. 3. **Identify sides relative to $\angle J$:** - Opposite side to $\angle J$ is $KI = 35$. - Adjacent side to $\angle J$ is $KJ = 12$. 4. **Calculate tangent:** $$\tan(\angle J) = \frac{\text{opposite}}{\text{adjacent}} = \frac{35}{12}$$ 5. **Simplify:** The fraction $\frac{35}{12}$ is already in simplest form. **Final answer:** $$\tan(\angle J) = \frac{35}{12}$$