To solve for $x$ in the given right triangle, we can apply the Pythagorean theorem.

Step-by-Step Solution

1. Identify the Triangle Components:

   • In the larger right triangle:
     • The hypotenuse is $17$.
     • The total base length is $8 + 4 = 12$.
   • The length $x$ is the height of the triangle.

2. Apply the Pythagorean Theorem: The Pythagorean theorem states: $\text{(Hypotenuse)}^2 = \text{(Base)}^2 + \text{(Height)}^2$ Plugging in the values: $17^2 = 12^2 + x^2$ $289 = 144 + x^2$

3. Solve for $x$: $x^2 = 289 - 144$ $x^2 = 145$ Taking the square root of both sides: $x = \sqrt{145}$ $x \approx 12.04$

Final Answer

$x \approx 12.04$

Would you like further details on any part of this solution?

Here are some related questions to consider:

1. What is the Pythagorean theorem, and how is it derived?
2. How do you identify which side is the hypotenuse in a right triangle?
3. Can the Pythagorean theorem be applied to non-right triangles?
4. How can the Pythagorean theorem be used in real-world applications?
5. What are the properties of a triangle with side lengths in the ratio of a Pythagorean triple?

Tip: When solving for a side in a right triangle, always double-check which side is the hypotenuse; it's the longest side opposite the right angle.