1. **Problem statement:** A right triangle has a horizontal leg measuring 42.7 cm and a hypotenuse measuring 52.5 cm. We need to find the length of the vertical leg. 2. **Formula used:** The Pythagorean theorem states that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse: $$a^2 + b^2 = c^2$$ where $a$ and $b$ are the legs and $c$ is the hypotenuse. 3. **Apply the formula:** Let the vertical leg be $a$, the horizontal leg be $b = 42.7$, and the hypotenuse be $c = 52.5$. $$a^2 + 42.7^2 = 52.5^2$$ 4. **Calculate squares:** $$a^2 + 1823.29 = 2756.25$$ 5. **Isolate $a^2$:** $$a^2 = 2756.25 - 1823.29$$ $$a^2 = 932.96$$ 6. **Find $a$ by taking the square root:** $$a = \sqrt{932.96}$$ $$a \approx 30.54$$ 7. **Answer:** The length of the vertical leg is approximately **30.54 cm**. This completes the solution for the first problem.