1. **Problem Statement:** We need to find the tangent of angle $X$ in a right triangle with sides $VW=77$, $WX=85$ (hypotenuse), and side $VX$ unknown. 2. **Formula:** The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle: $$\tan(X) = \frac{\text{opposite}}{\text{adjacent}}$$ 3. **Identify sides:** Angle $X$ is at vertex $X$. The side opposite $X$ is $VW=77$ (horizontal side), and the hypotenuse is $WX=85$. The side adjacent to $X$ is $VX$ (vertical side), which we need to find. 4. **Find the adjacent side $VX$ using the Pythagorean theorem:** $$VX = \sqrt{WX^2 - VW^2} = \sqrt{85^2 - 77^2}$$ Calculate inside the square root: $$85^2 = 7225, \quad 77^2 = 5929$$ So, $$VX = \sqrt{7225 - 5929} = \sqrt{1296} = 36$$ 5. **Calculate $\tan(X)$:** $$\tan(X) = \frac{\text{opposite}}{\text{adjacent}} = \frac{77}{36}$$ 6. **Simplify the fraction if possible:** 77 and 36 have no common factors other than 1, so the fraction is already in simplest form. **Final answer:** $$\boxed{\tan(X) = \frac{77}{36}}$$