1. **State the problem:** We have a right triangle with a hypotenuse of length 8.5, and the base is divided into two segments of lengths 3.2 and 6.8 by a height $x$ perpendicular to the base. 2. **Identify what to find:** We need to find the length of the height $x$. 3. **Recall the geometric property:** In a right triangle, the height to the hypotenuse relates the segments it divides by the formula: $$x = \sqrt{a \cdot b}$$ where $a$ and $b$ are the two segments of the hypotenuse. 4. **Apply the formula:** Here, $a = 3.2$ and $b = 6.8$, so $$x = \sqrt{3.2 \times 6.8}$$ 5. **Calculate the product:** $$3.2 \times 6.8 = 21.76$$ 6. **Calculate the square root:** $$x = \sqrt{21.76} = 4.664...$$ 7. **Final answer:** The height $x$ is approximately $$x \approx 4.66$$ This height divides the hypotenuse into two segments and is perpendicular to the base, confirming the right triangle properties.