To solve for $x$ in this image, let's analyze each triangle separately. This is a combination of two right triangles sharing a common vertical side.

1. Analyze the 30°-60°-90° Triangle (left triangle):

   • In a 30°-60°-90° triangle, the ratio of the sides opposite the 30°, 60°, and 90° angles is $1 : \sqrt{3} : 2$.
   • Here, the side adjacent to the 30° angle (20 m) represents the shorter leg of the triangle.

   Since the hypotenuse of this triangle will be twice the shorter leg: $\text{Hypotenuse} = 2 \times 20 = 40 \, \text{m}$

2. Calculate the vertical side (height) of the 30°-60°-90° Triangle:

   • This vertical side is opposite the 30° angle, so it is: $\text{Vertical side} = 20 \times \sqrt{3} = 20\sqrt{3} \, \text{m}$

3. Analyze the 45°-45°-90° Triangle (right triangle):

   • In a 45°-45°-90° triangle, the ratio of the sides is $1 : 1 : \sqrt{2}$.
   • The vertical side (height) of this triangle, which is also the height of the first triangle, is $20\sqrt{3}$.

   In a 45°-45°-90° triangle, the two legs are equal, so the other leg (horizontal side marked as $x$) is also $20\sqrt{3}$.

4. Calculate the value of $x$: $x = 20\sqrt{3} \approx 20 \times 1.732 = 34.64 \, \text{m}$

Thus, the value of $x$ is approximately 34.64 meters.

Would you like further details, or do you have any questions?

Here are some related questions to expand understanding:

1. How would the calculation change if the triangle had different angle values?
2. Can you verify this calculation using trigonometric functions for practice?
3. What are some other properties of 30°-60°-90° and 45°-45°-90° triangles?
4. How would you calculate the hypotenuse if only the vertical side was given?
5. How can we apply these triangle ratios to real-world problems?

Tip: In 30°-60°-90° and 45°-45°-90° triangles, knowing one side length often allows you to find all other sides quickly due to the fixed side ratios.