The problem shows a figure with two right triangles sharing a side, with the task being to solve for $x$, the hypotenuse of the top triangle.

Here’s how to approach the problem:

1. Break down the triangles:

   • The first triangle (top triangle) has legs of 5 and an unknown length, with $x$ as the hypotenuse.
   • The second triangle (bottom triangle) has legs of 6 and 9, forming a right triangle.

2. Solve the second (bottom) triangle:

   • Use the Pythagorean theorem: $c^2 = a^2 + b^2$ where $a = 6$, $b = 9$, and $c$ is the hypotenuse.

   $c = \sqrt{6^2 + 9^2} = \sqrt{36 + 81} = \sqrt{117} \approx 10.8$

3. Use the result for the shared side:

   • The shared side is a leg of the top triangle and equals 10.8.

4. Apply the Pythagorean theorem to the top triangle: $x^2 = 5^2 + 10.8^2$ $x^2 = 25 + 116.64 = 141.64$ $x = \sqrt{141.64} \approx 11.9$

Thus, $x$ to the nearest tenth is approximately 11.9.

Would you like further details or have any questions?

Related Questions:

1. How does the Pythagorean theorem apply to right triangles?
2. What are the properties of similar triangles, and how do they relate to this problem?
3. How can we calculate the length of the hypotenuse in different types of right triangles?
4. What other geometric methods can be used to solve for missing sides in complex figures?
5. How can trigonometric functions (like sine and cosine) be applied to find missing sides?

Tip:

Always ensure that the angles in a triangle add up to 180° and check the right-angle triangle conditions carefully.