1. **Problem statement:** Find the tangent ratios for angles $\angle Y$ and $\angle Z$ in a right triangle with vertices $X$, $Y$, and $Z$, where $X$ is the right angle. 2. **Given:** - Side $XY = 7$ - Side $XZ = 5$ - Hypotenuse $YZ = \sqrt{74}$ 3. **Recall the tangent ratio:** $$\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$$ 4. **Identify sides relative to each angle:** - For $\angle Y$: - Opposite side is $XZ = 5$ - Adjacent side is $XY = 7$ - For $\angle Z$: - Opposite side is $XY = 7$ - Adjacent side is $XZ = 5$ 5. **Calculate tangent ratios:** $$\tan Y = \frac{5}{7}$$ $$\tan Z = \frac{7}{5}$$ 6. **Answer:** The correct tangent ratios are $\tan Y = \frac{5}{7}$ and $\tan Z = \frac{7}{5}$, which corresponds to option (1).