1. **State the problem:** We have a right triangle with a base segment of 11.4 cm, an extended base segment of 7.1 cm, a hypotenuse of 12.2 cm, and an unknown height $x$. We need to find the length of $x$. 2. **Identify the right triangle sides:** The total base length is $11.4 + 7.1 = 18.5$ cm. 3. **Use the Pythagorean theorem:** For a right triangle with legs $a$ and $b$, and hypotenuse $c$, the formula is: $$a^2 + b^2 = c^2$$ Here, $a = x$ (height), $b = 11.4$ cm (base segment), and $c = 12.2$ cm (hypotenuse). 4. **Apply the formula:** $$x^2 + 11.4^2 = 12.2^2$$ Calculate squares: $$x^2 + 129.96 = 148.84$$ 5. **Isolate $x^2$:** $$x^2 = 148.84 - 129.96$$ $$x^2 = 18.88$$ 6. **Find $x$ by taking the square root:** $$x = \sqrt{18.88}$$ $$x \approx 4.34$$ 7. **Answer:** The height $x$ is approximately 4.34 cm. This completes the solution for the height $x$ in the right triangle.