To solve for $x$ using the given options, let's re-evaluate the problem and consider the correct usage of trigonometric functions in this context:

1. Identify the right triangle properties:

   • One angle is $64^\circ$, another is $27^\circ$, and the third angle is $90^\circ$.
   • The side adjacent to the $27^\circ$ angle is $97 + x$.
   • The side opposite the $27^\circ$ angle is the vertical side.

2. Using trigonometric relationships: Since we have the opposite and adjacent sides for angle $27^\circ$: $\tan(27^\circ) = \frac{\text{opposite}}{\text{adjacent}}$

3. Given $\tan(27^\circ) \approx 0.5095$: $\tan(27^\circ) = \frac{97 + x}{x}$ $0.5095 = \frac{97 + x}{x}$

4. Solve for $x$: $0.5095x = 97 + x$ $0.5095x - x = 97$ $-0.4905x = 97$ $x = \frac{97}{-0.4905}$ $x \approx -197.77$

Since the options provided are -10, 6, 10, and -6, let's use each to see which one fits:

1. Option $x = -10$: $0.5095 = \frac{97 - 10}{-10}$ $0.5095 \neq \frac{87}{-10} \Rightarrow 0.5095 \neq -8.7$

2. Option $x = 6$: $0.5095 = \frac{97 + 6}{6}$ $0.5095 = \frac{103}{6} \Rightarrow 0.5095 \neq 17.17$

3. Option $x = 10$: $0.5095 = \frac{97 + 10}{10}$ $0.5095 = \frac{107}{10} \Rightarrow 0.5095 \neq 10.7$

4. Option $x = -6$: $0.5095 = \frac{97 - 6}{-6}$ $0.5095 = \frac{91}{-6} \Rightarrow 0.5095 \neq -15.17$

Thus, none of the provided options (-10, 6, 10, -6) correctly satisfies the equation. There may be a need to re-evaluate the triangle setup or the problem's context.

Would you like more details, or do you have any questions? Here are 5 questions you might ask:

1. How can I verify the accuracy of my trigonometric calculations?
2. What are some common mistakes to avoid in trigonometry problems?
3. Can you provide a different method to solve this problem?
4. How do the given options affect the final solution?
5. Can you provide another example of solving a right triangle with different given sides?

Tip: Always double-check your trigonometric relationships and ensure your setup aligns with the given problem.