1. **State the problem:** We are given a right triangle with legs 6 cm and 10 cm, and we want to find the hypotenuse $x$ using the Pythagorean theorem. 2. **Formula:** The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: $$x^2 = a^2 + b^2$$ where $a$ and $b$ are the legs, and $x$ is the hypotenuse. 3. **Apply the formula:** Substitute $a=6$ and $b=10$: $$x^2 = 6^2 + 10^2$$ 4. **Calculate squares:** $$x^2 = 36 + 100$$ 5. **Sum the squares:** $$x^2 = 136$$ 6. **Solve for $x$ by taking the square root:** $$x = \sqrt{136}$$ 7. **Simplify the square root:** $$136 = 4 \times 34$$ $$x = \sqrt{4 \times 34} = \sqrt{4} \times \sqrt{34} = 2\sqrt{34}$$ **Final answer:** $$x = 2\sqrt{34} \approx 11.66$$ Note: The other expressions and graph mentioned are unrelated to this triangle problem and are not solved here.