To solve for $x$ in the given right triangle $\triangle NMO$, we will use trigonometric relationships. Here's how we proceed:

Step-by-Step Process:

1. Identify the triangle sides and angle:

   • $\angle NMO = 52^\circ$ (the given angle).
   • $NO = 7$ (the side opposite the angle $\angle NMO$).
   • $x = NM$ (the hypotenuse).

2. Select the trigonometric function: Since we have the opposite side ($NO$) and are looking for the hypotenuse ($NM$), we will use the sine function, which relates the opposite side and the hypotenuse.

   The sine function is given by: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ Substituting the known values: $\sin(52^\circ) = \frac{7}{x}$

3. Solve for $x$: $x = \frac{7}{\sin(52^\circ)}$ Using a calculator to find $\sin(52^\circ)$: $\sin(52^\circ) \approx 0.7880$ Therefore: $x = \frac{7}{0.7880} \approx 8.9$

So, the value of $x$ is approximately 8.9.

Would you like more details on any step or have further questions?

Here are 5 related questions to explore:

1. How do you determine which trigonometric function to use based on the sides of a triangle?
2. Can you use other trigonometric functions (cosine, tangent) to solve for different unknowns in this triangle?
3. How would you calculate the length of side $MO$ using trigonometry?
4. How do you check if a triangle is right-angled using side lengths?
5. How would the solution change if the angle given were different?

Tip: When solving trigonometric problems, always ensure your calculator is set to the correct mode (degrees or radians) depending on the angle given.