1. **Problem statement:** Find the value of $x$ in the first right triangle where one angle is $68^\circ$, the side adjacent to this angle is 38, and the side opposite is $x$. 2. **Formula and rules:** In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ 3. **Apply the formula:** Here, $\theta = 68^\circ$, opposite side = $x$, adjacent side = 38. $$\tan(68^\circ) = \frac{x}{38}$$ 4. **Solve for $x$:** Multiply both sides by 38: $$x = 38 \times \tan(68^\circ)$$ 5. **Calculate $\tan(68^\circ)$:** Using a calculator, $$\tan(68^\circ) \approx 2.4751$$ 6. **Find $x$:** $$x = 38 \times 2.4751 = 94.0538$$ 7. **Final answer:** $$x \approx 94.05$$ This means the length of the side opposite the $68^\circ$ angle is approximately 94.05 units.