1. **State the problem:** We have a right triangle with sides 14, 11, and $x$, where $x$ is the side opposite the right angle. We need to find the value of $x$. 2. **Formula used:** In a right triangle, the Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse (the side opposite the right angle), and $a$ and $b$ are the other two sides. 3. **Identify the hypotenuse:** The hypotenuse is the longest side. Between 14, 11, and $x$, 14 is the longest given side, so 14 is the hypotenuse. 4. **Apply the Pythagorean theorem:** $$11^2 + x^2 = 14^2$$ 5. **Calculate squares:** $$121 + x^2 = 196$$ 6. **Isolate $x^2$:** $$x^2 = 196 - 121$$ $$x^2 = 75$$ 7. **Find $x$ by taking the square root:** $$x = \sqrt{75}$$ 8. **Simplify the square root:** $$x = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$$ 9. **Approximate the value:** $$x \approx 5 \times 1.732 = 8.66$$ **Final answer:** $$x = 5\sqrt{3} \approx 8.66$$