To solve for $x$ in the triangle shown to the nearest tenth, we will apply the Pythagorean Theorem in the smaller right triangle and solve for the missing side.

Step-by-Step Solution:

1. Identify the right triangles:

   • The triangle with sides $4$, $x$ (hypotenuse), and the vertical right angle $4$ forms a smaller right triangle.

2. Use the Pythagorean Theorem for the smaller triangle: $a^2 + b^2 = c^2$ Here:

   • $a = 4$ (vertical side),
   • $b = 7$ (horizontal leg),
   • $c = x$ (hypotenuse of the smaller right triangle).

   Substituting the values: $4^2 + 7^2 = x^2$

3. Calculate: $16 + 49 = x^2$ $x^2 = 65$

4. Solve for $x$ by taking the square root: $x = \sqrt{65}$

5. Approximate to the nearest tenth: Using a calculator: $x \approx 8.1$

Final Answer:

The value of $x$ to the nearest tenth is 8.1.

Do you need further explanations or verification? Let me know if you need me to clarify any part of the solution.

Related Questions:

1. What is the Pythagorean Theorem and when is it used?
2. How do you determine which side of a triangle is the hypotenuse?
3. How can you approximate square roots without a calculator?
4. What are some real-world applications of right triangles and the Pythagorean Theorem?
5. How do you solve problems involving two right triangles within one figure?

Tip: Always identify the right triangle and label the sides carefully when solving geometric problems.