1. **State the problem:** We have a right triangle with sides 7.7, 11, and the hypotenuse labeled as $x$ and 10 (likely a typo, we consider $x$ as the hypotenuse). 2. **Identify the formula:** For a right triangle, the Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse. 3. **Assign values:** Let the legs be $7.7$ and $11$, and the hypotenuse be $x$. 4. **Apply the formula:** $$7.7^2 + 11^2 = x^2$$ 5. **Calculate squares:** $$7.7^2 = 59.29$$ $$11^2 = 121$$ 6. **Sum the squares:** $$59.29 + 121 = 180.29$$ 7. **Solve for $x$:** $$x^2 = 180.29$$ $$x = \sqrt{180.29}$$ 8. **Calculate the square root:** $$x \approx 13.42$$ **Final answer:** $$x \approx 13.42$$